Generalized sampling and the stable and accurate

Generalized sampling and the stable and accurate
Generalized sampling and the stable and accurate
reconstruction of piecewise analytic functions from their
Fourier coefficients
Ben Adcock
Department of Mathematics
Simon Fraser University
Burnaby, BC V5A 1S6
Canada
Anders C. Hansen
DAMTP, Centre for Mathematical Sciences
University of Cambridge
Wilberforce Rd, Cambridge CB3 0WA
United Kingdom
November 21, 2011
Abstract
Suppose that the first m Fourier coefficients of a piecewise analytic function are given. Direct expansion in a Fourier series suffers from the Gibbs phenomenon and lacks uniform convergence. Nonetheless,
in this paper we show that, under very broad conditions, it is always possible to recover an n-term expansion in a different system of functions using only these coefficients. Such an expansion can be made
arbitrarily close to the best possible n-term expansion in the given system. Thus, if a piecewise polynomial
basis is employed, for example, exponential convergence can be restored. The resulting method is linear,
numerically stable and can be implemented efficiently in only O (nm) operations.
A key issue is how the parameter m must scale in comparison to n to ensure recovery. We derive
analytical estimates for this scaling for large classes of polynomial and piecewise polynomial bases. In
particular, we show that in many important
cases, including the case of piecewise Chebyshev polynomials,
this scaling is quadratic: m = O n2 . Therefore, with a system of polynomials that the user is essentially free to choose, one can restore exponential accuracy in n and root exponential accuracy in m. This
generalizes a result proved recently for piecewise Legendre polynomials.
The method developed in this paper is part of new numerical framework for sampling and reconstruction in abstract Hilbert spaces, known as generalized sampling. This paper extends previous work by the
authors by introducing a substantially more flexible framework which allows for sampling and reconstruction with respect to different inner products. In the final part of this paper we illustrate the application of
generalized sampling to a related family of problems.
1
Introduction
The Fourier expansion of an analytic and periodic function f converges exponentially fast in the truncation
parameter m. For this reason – as well as the fact that the expansion can be computed rapidly via the Fast
Fourier Transform (FFT) – such approximations are extremely widely used in all areas of computational
mathematics, with one important instance being the spectral solution of partial differential equations (PDEs)
[15, 19]. However, rapid convergence is lost when the function is no longer analytic. Indeed, when f is only
piecewise analytic, one encounters the familiar Gibbs phenomenon. This not only destroys local accuracy
– characteristic O (1) oscillations are witnessed near each jump in the function – it also inhibits global
approximation: the truncated expansion converges only very slowly in the L2 norm, and not at all when the
error is measured in the uniform norm. Notably, this issue arises when computing spectral approximations
to PDEs that develop shocks – hyperbolic conservation laws, for example [31, 41].
This naturally leads to the following question: given the first m Fourier coefficients of a piecewise
analytic function f , how can one recover high orders of approximation? The problem is not new, and there
have been many different approaches developed for its solution (see, for example, [2, 11, 12, 17, 18, 22,
23, 24, 25, 52] and references therein). Of this large collection, perhaps the most successful and commonly
used is the method of spectral reprojection [30, 33, 34, 35]. Introduced by D. Gottlieb et al [35], in this
approach the Fourier coefficients of f are used to approximate the first n = O (m) coefficients in a new
1
basis of piecewise polynomials (the reconstruction basis). Provided this basis is chosen appropriately, one
obtains exponential accuracy in m. To date, spectral reprojection has been successfully used in a range of
applications, including image reconstruction [9, 10], and the postprocessing of spectral discretizations of
hyperbolic PDEs [28, 31, 41].
Whilst spectral reprojection has been widely successful, there are a number of drawbacks. In particular,
spectral reprojection achieves rapid convergence by employing a particular choice of reconstruction basis.
Herein lies a problem. Only very few bases (known as Gibbs complementary bases [30, 34]) have this
property, with the two most commonly used being ultraspherical (Gegenbauer) and Freud polynomials. In
both cases, however, the parameter α defining the polynomials must scale linearly with n. This is not
only computationally inconvenient – changing n requires recomputation of the whole reprojection basis
– in addition, a rather careful selection of parameters must be employed to ensure such convergence and
avoid potentially a Runge-type phenomenon [16]. Because of the stringent requirements placed on the
reconstruction basis, spectral reprojection consequently affords little flexibility to the user. At the very
least, Gegenbauer and Freud polynomials are no where near as easy to use and manipulate as Chebyshev
polynomials, for example, with the latter being amenable to the FFT. Another issue, as shown empirically in
[7], is that, even if exponential convergence occurs, the rate of this convergence in the Gibbs complementary
basis may be quite slow in practice, meaning that many Fourier coefficients may be required to achieve high
accuracy.
With this in mind, the purpose of this paper is to consider a different approach based on an alternative
idea. Unlike spectral reprojection, where the reconstruction basis must be suitably chosen to ensure rapid
convergence, the method we develop in this paper allows the user to employ an arbitrary basis. In other
words, we address the following problem:
Problem 1.1. Given the first m Fourier coefficients of a piecewise analytic function f , where m is sufficiently large, recover an approximation fn,m to the n-term expansion Qn f in an arbitrary basis of piecewise polynomials (the reconstruction basis) orthogonal to with respect the inner product h·, ·i satisfying
kf − fn,m k kf − Qn f k as n, m → ∞, where k·k is the corresponding norm corresponding.
∞
Here and elsewhere we use the notation an bn for nonnegative sequences {an }∞
n=1 and {bn }n=1 to
mean that, for all sufficiently large n there exist c1 , c2 > 0 such that c1 an ≤ bn ≤ c2 an . Observe that
kf − fn,m k kf − Qn f k implies that the approximation fn,m is quasi-optimal to f from the set Pn : as
n → ∞, fn,m converges to f at precisely the same rate as the best approximation Qn f .
As we show, by letting m vary independently of n, the restriction of having to employ a Gibbs complementary basis is avoided. This allows one to use far more convenient polynomial bases in reconstruction,
such as the aforementioned Chebyshev basis. Note that the case of (piecewise) Legendre polynomial bases
was developed in [7] (therein several examples demonstrating improved performance over spectral reprojection were also given). The main contribution of this paper (see §1.4 for a more detailed description) is to
generalize this work to arbitrary bases of piecewise polynomials.
1.1
Reconstructions in Hilbert spaces
Our solution to Problem 1.1 is based on recasting it in terms of sampling and reconstruction in abstract
Hilbert spaces. To this end, let HS and HR be subspaces of a vector space V that form Hilbert spaces with
∞
respect to the bilinear forms h·, ·iS and h·, ·iR . Suppose that {ψj }∞
j=1 , {φj }j=1 are orthonormal bases for
S
R
(H , h·, ·iS ) and (H , h·, ·iR ) respectively (the sampling and reconstruction bases), and let U ⊆ HR be a
subspace (U will consist of those functions f we wish to reconstruct). For f ∈ U, let
fˆj = hf, ψj iS ,
j = 1, 2, . . . ,
be the samples of f . The method we develop in this paper is designed to solve the problem:
Problem 1.2. Given the first m samples fˆ1 , . . . , fˆm of f ∈ U, recover an approximation fn,m to Qn f
satisfying kf − fn,m kR kf − Qn f kR , where Qn : (HR , h·, ·iR ) → Tn := span{φ1 , . . . , φn } is the
orthogonal projection.
It transpires that, under certain assumptions, this problem is solvable, and the resulting framework is
linear, and completely numerically stable. The key is that m must scale appropriately to n. The required
scaling can be derived numerically, and, in the cases we consider in this paper, analytically.
2
It is straightforward to see that this framework can be applied to Problem 1.1. In particular, consider
the simple scenario where f is analytic, but nonperiodic (equivalently, f : T → R has a jump at x = −1,
where T = [−1, 1) is the unit torus). Given the Fourier coefficients of f , we would like to reconstruct in the
orthonormal basis of Chebyshev polynomials φj (x) = cj Tj (x) of the first kind (for example), where
Tj (x) = cos(j arccos x),
and c0 =
√1 ,
π
cj =
q
2
π , j 6=
2 − 21
j = 0, 1, 2, . . . ,
1. Given that the polynomials Tj are orthogonal with respect to the weight
function w(x) = (1 − x ) , we thus let HS = L2 (−1, 1) and HR = L2w (−1, 1) (the spaces of squareintegrable functions and weight square-integrable functions with respect to the weight function w respectively), and define ψj (x) = √12 exp(ijπx) to be the standard Fourier basis.
As is no doubt apparent to the reader, Problem 1.2 is actually far more general than Problem 1.1. Indeed,
the abstract framework introduced to solve Problem 1.2 can be applied far more widely than the problem of
reconstructing from Fourier data. We shall discuss several other important applications of this framework at
the end of this paper. In particular, we illustrate how this framework can be used to solve another common
problem of this type: namely, the reconstruction of a piecewise analytic function from its coefficients with
respect to an orthogonal polynomial basis. Much like the Fourier case, the problem also notably occurs in
spectral discretizations of hyperbolic PDEs [31, 34, 41].
1.2
Orthogonal polynomial systems
It is useful at this moment to introduce some notation. We shall be principally concerned in this paper with
Jacobi polynomials, with corresponding weight function
wα,β (x) = (1 − x)α (1 + x)β ,
α, β > −1.
We write L2α,β (−1, 1) for the space of weighted square-integrable functions with respect to wα,β , and k·kα,β
for the corresponding norm. Our main examples herein will be ultraspherical (or Gegenbauer) polynomials
(α = β = γ), and in particular, Legendre polynomials (γ = 0) and Chebyshev polynomials of the first
(γ = − 21 ) and second (γ = 12 ) kinds. Whenever α = β = γ we shall use the slightly more succinct notation
L2γ (−1, 1), k·kγ and wγ .
Legendre and Chebyshev polynomials are the most common in applications – in particular, spectral methods for PDEs [15, 19]. More specifically, Chebyshev polynomials are chosen for reconstruction because of
their aforementioned computational flexibility, whereas Legendre polynomials are desirable because of the
simplicity of the Legendre weight function w(x) ≡ 1. However, it is of both theoretical and practical interest
to maintain the generality of Jacobi polynomials. Indeed, whilst Legendre and Chebyshev polynomials are
most frequently used in practice, there are a number of applications which employ more general ultraspherical and Jacobi polynomial systems [15, 19, 36, 37].
If f is piecewise analytic with jumps at −1 < x1 < . . . < xl < 1 we shall seek to reconstruct in bases of
piecewise polynomials. Thus, with α = {α0 , . . . , αl } and β = {β0 , . . . , βl } we define the piecewise Jacobi
weight function wα,β (x) by
wα,β (x) = (xr+1 − x)αr (x − xr )βr ,
x ∈ Ir := (xr , xr+1 ),
r = 0, . . . , l.
Here x0 = −1 and xl+1 = 1. The corresponding orthonormal system of piecewise polynomials can be
obtained by appropriately scaling from the standard interval [−1, 1] to each subinterval of smoothness Ir
(see §3.1). When considering a finite expansion in such functions, we shall write n = (n0 , . . . , nl ) ∈ Nl+1
for the vector of polynomial degrees in each subinterval Ir , and define
Tn = {φ : φ|Ir ∈ Pnr , r = 0, . . . , l} .
(1.1)
The operator Qn : L2α,β (−1, 1) → Tn will be the orthogonal projection with respect to the weight function
wα,β . As in the no-jump (analytic and nonperiodic) setting, the main examples we consider in this paper
involve reconstructions in piecewise Chebyshev or Legendre polynomials, i.e. α = β = γ, where γ =
{− 21 , . . . , − 12 } (first kind), γ = { 12 , . . . , 21 } (second kind) or γ = {0, . . . , 0} respectively.
Remark 1.3 Many of the results we prove in this paper extend quite trivially to the modified Jacobi weight
wα,β (x) = g(x)(1 − x)α (1 + x)β ,
3
α, β > −1,
(1.2)
where g(x) is analytic and positive on [−1, 1]. Likewise, one may define the piecewise modified Jacobi
weight
(1.3)
wα,β (x) = gr (x)(xr+1 − x)αr (x − xr )βr , x ∈ Ir := (xr , xr+1 ),
where gr is analytic and positive in Ir . Although wα,β , as given by (1.3), is only unique (for given α and
β) up to multiplication by a collection g0 , . . . , gl of positive, analytic functions, we shall continue to write
L2α,β (−1, 1) for the corresponding space of square-integrable functions with respect to (1.3), and shall not
make the dependence on the functions g0 , . . . , gl explicit.
1.3
Key results
We devote the first part of this paper to the solution of Problem 1.2 and the properties of the resulting
framework, including numerical stability. In the second half, we consider its application to Problem 1.1. In
this regard, the key result we prove is as follows:
Theorem 1.4. Let α = {α0 , . . . , αl } and β = {β0 , . . . , βl }, where αr , βr > −1, r = 0, . . . , l, be parameters corresponding to the piecewise (modified) Jacobi weight wα,β , and suppose that Qn is the orthogonal
projection onto Tn , as given by (1.1), with respect to wα,β . Define
αr
−1 < αr < 1
βr
−1 < βr < 1
α̃r =
,
β̃r =
,
1 − r
αr ≥ 1
1 − 0r
βr ≥ 1
where 0 < r , 0r < 2 are arbitrary, and let U := L2α̃,β̃ (−1, 1). Suppose that the first m Fourier coefficients
of f ∈ U are given. Then there exists a c > 0 independent of m, n = (n0 , . . . , nl ) and f such that, whenever
m ≥ cn̄r , where n̄ = max{n0 , . . . , nl } and
p−q
,
r = max 2, p + 1, 2 + 2
q+1
p = max{α0 , . . . , αl , β0 , . . . , βl },
q = min{α0 , . . . , αl , β0 , . . . , βl },
one may compute, in a completely numerically stable manner, a reconstruction fn,m ∈ Tn satisfying
kf − Qn f kα,β ≤ kf − fn,m kα,β ≤ c0 kf − Qn f kα̃,β̃ ,
for some c0 > 0 depending only on c. Moreover, excluding a preprocessing step of at most O m2 cost,
the number of operations required to compute fn,m is O (nm). If wα,β is a piecewise ultraspherical weight,
then such preprocessing involves only O (nm) operations.
This theorem guarantees solution of Problem 1.1 for a very large class of functions. In fact, if −1 <
αr , βr < 1, r = 0, . . . , l, then we can recover any f ∈ U ≡ HR , where HR = L2α,β (−1, 1). In the general
case, note that U, although not equal to HR , is still a very large space, and is therefore certainly sufficient
for practical purposes (recall that the principal concern of this paper is piecewise analytic functions). Of
course, higher smoothness of f ensures fast convergence of Qn f , and therefore fn,m . However, this theorem
demonstrates that quasi-optimal, stable recovery with GS is always possible regardless of smoothness.
In practice, it is important to understand the constant c of Theorem 1.4. Fortunately, c can always
be computed numerically (see §2.3). Also, empirical evidence suggests that a small value of c is usually
suitable. In the numerical experiments in this paper we use c = 41 , which gives perfectly acceptable results
(see §4). One could also use a smaller value, with the only difference being a slightly larger, but still O (1),
condition number.
The proof of Theorem 1.4 relies on careful analysis of the behaviour of Fourier expansions in certain
weighted spaces. Herein, a connection is made with the Helson–Szegö Theorem on positive measures [26].
Another feature of our analysis is that, in many cases, the best constants corresponding to the quasi-optimality
kf − fn,m kR kf − Qn f kR are known, and can be explicitly computed. Specifically, one has the sharp
bounds
1
kf − Qn f kR ,
kf − fn,m kR ≤
Cn,m
where the constant Cn,m is bounded (provided m ≥ cn̄r ) and has a geometrical interpretation in terms of
the angle between two particular finite-dimensional subspaces.
4
Note also the great flexibility guaranteed by this theorem. In many important cases, m = O n2 samples
of a piecewise analytic function allow one to recover close to the best n-term approximation in a given
piecewise polynomial system. This includes (piecewise) Legendre (p = q = 0) and Chebyshev polynomials
(p = q = ± 21 ), both of which are commonly used in applications. In particular,
Corollary 1.5. Given the first m Fourier coefficients of a piecewise analytic function f , one can compute, in
3
O(m 2 ) operations, a root-exponentially convergent piecewise polynomial approximation of f that consists
of arbitrary piecwise ultraspherical polynomials with parameter −1 < γ < 1.
In summary, the method we present in this paper improves on the more conventional approach of spectral
reprojection by allowing reconstructions
in arbitrary polynomial bases. Whatsmore, spectral reprojection
incurs a relatively large cost of O m2 operations. Here we also obtain an improvement, by a factor of
1
O(m 2 ). Having said this, spectral reprojection is formally exponentially convergent, whereas this approach
obtains only root-exponential convergence. However, as we shall demonstrate by numerical example (see
§4.2), this rate of exponential convergence may be quite slow in practice, meaning that the approach of
this paper, whilst formally less rapidly convergent, actually gives a better numerical approximation in many
cases.
Remark 1.6 At first sight it may appear that the scaling m = O n2 , resulting in root exponential convergence in m, is less than optimal, and therefore could be improved. As discussed in [6] (see also [50] for a
closely related result), root-exponential convergence is the best that can be attained by a stable method for
this problem, and thus the framework developed in this paper is optimal in this sense.
1.4
Relation to previous work
The special case HS = HR = H, h·, ·iR = h·, ·iS = h·, ·i was first considered in [4, 7]. Therein an abstract
framework, known as generalized sampling (GS), was developed to solve Problem 1.2 in this more simple
setting. The framework introduced in this paper to address the significantly more general case HS 6= HR ,
h·, ·iS 6= h·, ·iR is a direct extension of this work, and, for this reason, we shall continue to refer to the
resulting framework as generalized sampling.
The original GS framework of [4, 7] can be applied successfully to Problem 1.1 whenever Qn f is the
Legendre polynomial expansion of f (this corresponds to HR = HS = L2 (−1, 1)), leading to a stable
numerical method. This particular instance of GS was also is described [42]. Unfortunately this framework
is not as flexible as one may hope. Although one can also reconstruct using arbitrary bases of polynomials,
the resulting method is only numerically stable when Legendre polynomials are employed. Moreover, in the
3
ultraspherical case, for example, this method also incurs an increased cost (over the value of O(m 2 ) for the
3+|γ|
Legendre case) proportional to O(m 2 ), where γ is the corresponding polynomial parameter.
Reconstructions in arbitrary orthogonal polynomial systems from Fourier coefficients have also been
considered in [44, 45]. However, the approach introduced therein is exponentially unstable, and liable to
suffer from a Runge-type phenomenon (i.e. divergence for insufficiently analytic functions). The work of
[42] can be viewed as a regularization of this approach, yet only the case of Legendre polynomials was
considered. Fortunately, the extended GS framework we develop in this paper, which exploits the orthogonality of both the sampling and reconstruction bases, allows for completely stable reconstructions in only
3
O(m 2 ) operations using significantly more general families of polynomials. Moreover, as we explain in
due course, the extended GS framework can actually be reinterpreted as a preconditioning of the original GS
framework. This preconditioner acts to stabilize the particular linear system, and thereby reduces the overall
computational cost.
1.5
Outline
The outline of the remainder of this paper is as follows. In §2 we introduce and analyze the extended version
of the GS framework to solve Problem 1.2. §3 is devoted to (piecewise) polynomial reconstructions from
Fourier samples, and in particular, the solution of Problem 1.1 and proof of Theorem 1.4. In §4 we give
numerical examples and in §5 we discuss applications of this extended GS framework to several related
reconstruction problems.
5
2
An extended generalized sampling framework
Let HS and HR be subspaces of a vector space V that form Hilbert spaces in their own right with respect to
the bilinear forms h·, ·iS and h·, ·iR respectively. Let U ⊆ HR be a subspace (not necessarily closed) and
R
assume that f ∈ U and that {φj }∞
j=1 ⊆ U, where {φj } is some orthonormal basis for (H , h·, ·iR ). The
subspace U consists of those functions f that we wish to recover and the subspace Tn := span{φ1 , . . . , φn }
S
is the space in which we seek to reconstruct. If {ψj }∞
j=1 ⊆ U is an orthonormal basis of (H , h·, ·iS ), we
assume that we have access to the first m samples of f
fˆj = hf, ψj iS ,
j = 1, . . . , m,
(2.1)
with respect to this basis (we shall assume that V is such that these values exist and are finite). In practice, the
basis {φj }∞
j=1 is chosen with some a priori knowledge about f . For the applications considered in this paper,
where f is a piecewise analytic function on [−1, 1], {φj }∞
j=1 will consist of orthonormal piecewise polynomials with respect to some weight function and ψj will correspond to the complex exponential √12 eijπx (in
this case we shall enumerate ψj over Z as opposed to N).
Remark 2.1 Note that the assumption of orthonormality in the sampling and reconstruction vectors {ψj }∞
j=1 ,
is
not
necessary.
It
is
reasonably
straightforward
to
relax
this
condition
to
that
of
a
Riesz
basis
[7]
{φj }∞
j=1
(and also further to a frame [8]). However, since all sampling and reconstruction vectors of interest in this
paper will be orthonormal, we shall accept this slight loss of generality.
We seek to solve Problem 1.2: namely, compute an approximation fn,m ∈ Tn from the samples (2.1)
that is close to Qn f , where Qn : (HR , h·, ·iR ) → Tn is the orthogonal projection. Note that Qn f is defined
by the equations
hQn f, φj iR = hf, φj iR , j = 1, . . . , n, Qn f ∈ Tn .
(2.2)
Had we had access to the values hf, φj iR , i.e. the coefficients of f in a particular polynomial basis, for
example, we could have computed Qn f immediately. However, in general we do not have this information.
Specifically, all we know is {fˆj }m
j=1 . Since the sampling basis {ψj }j∈N is fixed, and cannot be altered, we
are unable to form Qn f from the given data.
Nonetheless, let us write Pm : HS → Sm = span{ψ1 , . . . , ψm } for orthogonal projection onto Sm ⊆ U
with respect to h·, ·iS , i.e.
m
X
Pm g =
hg, ψj iS ψj , g ∈ HS .
j=1
We now define fn,m ∈ Tn as the solution to
hPm fn,m , Pm φiR = hPm f, Pm φiR ,
∀φ ∈ Tn ,
fn,m ∈ Tn .
(2.3)
Suppose for a moment that (HS , h·, ·iS ) and (HR , h·, ·iR ) coincide. This is precisely the setting of the original
GS framework of [7]. It is now quite simple to give an intuitive explanation as to how fn,m solves Problem
S
1.2. Indeed, since {ψj }∞
j=1 is an orthonormal basis for (H , h·, ·iS ), we have that Pm → I in the strong
S
S
S
operator topology on (H , h·, ·iS ) [20], where I : H → H is the identity operator. Thus, for fixed n ∈ N,
if m → ∞ the equations (2.3) resemble those equations (2.2) defining Qn f . Therefore, for sufficiently large
m, we expect fn,m to inherit the features of Qn f (in particular, fn,m should exist uniquely), and critically
its good approximation properties.
In the general case (HS , h·, ·iS ) 6= (HR , h·, ·iR ), we need the additional assumption that Pm → I
strongly on (U, h·, ·iR ):
kg − Pm gkR → 0, m → ∞, ∀g ∈ U.
(A1)
Note that this assumption is not guaranteed in general, and must be verified for each particular problem.
However, as we shall see in §3, it does hold in all cases of interest in this paper. We will also require one
further assumption: namely, a uniform boundedness condition for {Pm }m∈N in terms of some norm ||| · |||
defined on U:
kPm gkR ≤ c|||g|||, ∀g ∈ U, m ∈ N,
(A2)
where c > 0 is independent of m and g.
6
In some circumstances, but not all, it transpires that the following stronger version of (A2) holds:
U = HR ,
Pm : HR , h·, ·iR → HR , h·, ·iR bounded.
(A20 )
To see that this is stronger than (A2), we note that
Lemma 2.2. Suppose that (A1) and (A20 ) hold. Then the family {Pm }m∈N is uniformly bounded. In other
words, (A2) holds with ||| · ||| = k·kR .
Proof. (A1) and (A20 ) imply that supm∈N kPm gk < ∞ for any g ∈ U = HR . The uniform boundedness
principle now gives the result.
Whenever (A20 ) holds instead of (A2) we may give a more precise analysis of the approximation fn,m
(see §2.2). In §3 we detail the situations in which (A20 ) holds as opposed to (A2) for the Fourier reconstruction problem.
2.1
Recovery from Fourier coefficients
Before analysing this framework, let us first discuss the main example of this paper. Suppose that f :
[−1, 1] → R is piecewise analytic with jumps at −1 < x1 < . . . < xl < 1. Let ψj (x) = √12 exp(ijπx),
j ∈ Z, and assume that the coefficients
Z 1
jmk
jmk
ˆ
fj =
,...,
,
(2.4)
f (x)ψj (x) dx, j = −
2
2
−1
are given. In particular, Pm f is the truncated Fourier series of f (we henceforth refer to Pm as the Fourier
projection operator in this case). We wish to recover f in a basis of orthonormal piecewise polynomials
corresponding to the piecewise (modified) Jacobi weight wα,β given by (1.3), where α = {α0 , . . . , αl } and
β = {β0 , . . . , βl }. To this end, let x0 = −1, xl+1 = 1 and write
Tn = {φ : φ|Ir ∈ Pnr : r = 0, . . . , l} ,
n = (n0 , . . . , nl ),
(2.5)
where Ir = (xr , xr+1 ). We now compute fn,m ∈ Tn using (2.3). Thus, we let V = L1 (−1, 1) (the space of
absolutely integrable functions), HR = L2α,β (−1, 1), k·kR = k·kα,β , and set HS = L2 (−1, 1), k·kS = k·k.
There are now three key questions:
(i) How do we choose the subspace U and the norm ||| · ||| so that (A1) and (A2) hold?
(ii) Does such a choice of U include all functions of interest? Specifically, U should at the very least
contain all piecewise analytic functions.
(iii) In what circumstances does (A20 ) hold instead of (A2)?
In §3 we provide answers to these questions.
Remark 2.3 Throughout we assume that the discontinuity locations x1 , . . . , xl are known. Indeed, this is
necessary prior knowledge for constructing an appropriate reconstruction basis of piecewise polynomials. In
practice, a fully-automated algorithm must also incorporate a scheme for locating such singularities. There
are numerous methods for this problem, and we shall not discuss this issue any further. We refer the reader
to [29, 52].
2.2
Analysis of the extended framework
We present two types of analysis of this framework. The first, for which the corresponding error estimates
(i.e. bounds for kf − fn,m kR ) are not sharp, is valid whenever (A1) and (A2) hold. The second, which
assumes the stronger condition (A20 ), leads to sharp bounds.
7
2.2.1
Version I
The definition (2.3) of fn,m motivates the introduction of the quantity
En,m = sup {kφ − Pm φkR : φ ∈ Tn , kφkR = 1} .
(2.6)
This quantity measures how close the restriction Pm |Tn is to the identity operator I|Tn . Since (A1) holds
and Tn is finite-dimensional, the following lemma is unsurprising:
Lemma 2.4. Suppose that En,m is defined by (2.6). Then En,m ≤ 1 + cdn for all m, n ∈ N, where c is as
in (A2) and
dn = sup {|||φ||| : φ ∈ Tn , kφkR = 1} .
(2.7)
Moreover, En,m → 0 as m → ∞ for fixed n.
Proof. Note that En,m ≤ 1 + sup {kPm φkR : φ ∈ Tn , kφkR = 1}. (A2) now gives the first result. For the
second, observe that the set Bn := {φ ∈ Tn : kφkR = 1} is compact. Hence, since Pm → I strongly on
U ⊇ Bn , this convergence is uniform on Bn .
With this to hand, we are now able to prove the main result of this section:
Theorem 2.5. For every n ∈ N there exists an m0 such that the approximation fn,m , defined by (2.3), exists
and is unique and satisfies the estimates
kfn,m kR ≤
c
|||f |||,
1 − En,m
(2.8)
and
kf − fn,m kR ≤ kf − Qn f kR +
1
cEn,m
|||f − Qn f ||| +
k(I − Pm )(f − Qn f )kR , (2.9)
(1 − En,m )2
(1 − En,m )2
where Qn : HR → Tn is the orthogonal projection with respect to h·, ·iR and En,m is as in (2.6). Specifically, m0 is the least m such that En,m < 1.
Note that (2.8) is a continuous stabilty estimate for fn,m . In particular, this implies that the coefficients
of fn,m cannot grow large.
Proof. Let U : Tn → Cn be the linear operator g 7→ {hPm g, Pm φj iR }nj=1 . It suffices to show that U
is invertible; in other words, ker(U) = {0}. Suppose that g ∈ Tn with Ug = 0. Then, by linearity,
hPm g, Pm φiR = 0, ∀φ ∈ Tn . In particular, 0 = kPm gkR ≥ (1 − En,m )kgkR . Thus, if m ≥ m0 , where
m0 is the least m such that En,m < 1, then we must have that g = 0. Hence, U is nonsingular, and fn,m
exists uniquely.
Now consider (2.8). Setting φ = fn,m in (2.3) gives
kPm fn,m k2R = hPm f, Pm fn,m iR ≤ kPm f kR kPm fn,m kR ≤ c|||f |||kPm fn,m kR .
The inequality (2.8) now follows directly from the definition of En,m and (A2).
For (2.9) we first write e = fn,m − Qn f ∈ Tn and notice that
kPm ek2R = hPm (f − Qn f ), Pm eiR .
Since e ∈ Tn and f − Qn f ⊥ Tn is the orthogonal projection, the right hand side may be written as
hPm (f − Qn f ), Pm eiR −hf − Qn f, eiR
= −h(I − Pm )(f − Qn f ), eiR − hPm (f − Qn f ), (I − Pm )eiR .
This gives
(1 − En,m )2 kek2R ≤ kPm ek2R ≤ k(I − Pm )(f − Qn f )kR kekR + kPm (f − Qn f )kR k(I − Pm )ekR .
Hence, by (A2) and the definition of En,m , we obtain
(1 − En,m )2 kekR ≤ k(I − Pm )(f − Qn f )kR + cEn,m |||f − Qn f |||.
The full result follows from the inequality kf − fn,m kR ≤ kekR + kf − Qn f kR .
8
This theorem confirms solution of Problem 1.2 using (2.3) whenever (A1) and (A2) hold, and provided
m is sufficiently large in comparison to n. The question of how large will be discussed and quantified in
§2.3. Observe that
kg − Pm gkR ≤ kgkR + c|||g|||, ∀g ∈ U, ∀m ∈ N,
hence the term k(I − Pm )(f − Qn f )kR in right-hand side of (2.9) can be easily bounded independently of
m to give the more convenient bound
"
#
1
c(1 + En,m )
|||f − Qn f |||.
(2.10)
kf − fn,m kR ≤ 1 +
kf − Qn f kR +
2
(1 − En,m )
(1 − En,m )2
However, this bound obscures one interesting facet of fn,m : namely, its asymptotic optimality. That is, by
increasing the number of samples m, fn,m can be made arbitrarily close to the best approximation Qn f .
Indeed,
Corollary 2.6. Suppose that fn,m is defined by (2.3). Then, for fixed n, fn,m → Qn f as m → ∞.
Proof. Since En,m → 0 as m → ∞ (Lemma 2.4), we have
kekR ≤
1
cEn,m
|||f − Qn f ||| +
k(I − Pm )(f − Qn f )kR → 0,
2
(1 − En,m )
(1 − En,m )2
as m → ∞, where e = fn,m − Qn f . Thus, fn,m → Qn f as required.
The analysis of this section demonstrates quasi-optimality of fn,m (as well as asymptotic optimality).
However, it is clearly less than ideal to have an error estimate of the form (2.9) involving |||f − Qn f |||. In
general, there is no guarantee that this term decays as n → ∞ (although this is always true in the applications
of this paper). Thus, one may ask: can a better analysis give an error bound involving only kf − Qn f kR
(which must tend to zero since Qn : HR → Tn is the orthogonal projection)? As it transpires, whenever
(A20 ) holds instead of (A2), this is indeed the case. This is described in the next section. The resulting
analysis also provides bounds for kfn,m kR and kf − fn,m kR which, unlike those given in Theorem 2.5, are
sharp.
2.2.2
Version II
Suppose now that (A20 ) holds instead of (A2). To derive an improved analysis, it is useful to introduce the
notion of oblique projections in Hilbert spaces:
Definition 2.7. Let U and V be closed subspaces of a Hilbert space H. A mapping W := WUV : H → U is
an oblique projection onto U along V if W 2 = W and W(v) = 0, ∀v ∈ V.
We shall also require the following:
Definition 2.8. Let U and V be closed subspaces of a Hilbert space H and QV : H → V the orthogonal
projection onto V. The subspace angle θ = θUV between U and V is given by
cos(θUV ) = inf {kQV uk : u ∈ U, kuk = 1} .
Note that, U ⊕ V = H if and only if there exists a unique oblique projection onto U along V [51].
Equivalently, the cosine of the subspace angle cos θUV⊥ 6= 0. In the particular case that V = U⊥ (i.e.
cos θUV = 1), WUV = QU is the orthogonal projection onto U.
We now require the following theorem:
Theorem 2.9 ([6]). Suppose that U and V are closed subspaces of H satisfying U ⊕ V⊥ = H, and let
W : H → U be the oblique projection onto U along V⊥ . Then we have the sharp bounds
kWf k ≤
1
kf k,
cos θUV
∀f ∈ H,
(2.11)
and
1
kf − Qf k, ∀f ∈ H,
cos θUV
where Q : H → U is the orthogonal projection. In particular, if f ∈ U then Wf = f .
kf − Qf k ≤ kf − Wf k ≤
9
(2.12)
0
This aside, note that,
the operator Pm is bounded on HR . Hence it has a bounded
since R(A2 ) holds,
∗
R
∗
adjoint Pm : H , h·, ·i → H , h·, ·i . Write Wm := Pm
◦ Pm . We require the following lemma:
Lemma 2.10. The operators Wm converge strongly to I on HR , h·, ·i .
Proof. For g ∈ HR , we have
∗
∗
kg − Wm gk2 = hg − Pm
Pm g, g − Pm
Pm gi
∗
∗
∗
∗
≤ |hPm
(g − Pm g), g − Pm
Pm gi| + |hg − Pm
g, g − Pm
Pm gi|.
∗
Since {Pm }m∈N is uniformly bounded, so is {Pm
}m∈N . Moreover, the adjoint operation is strongly contin∗
uous on bounded sets. Hence Pm → I strongly on HR and we obtain, for some c > 0 independent of g and
m,
∗
kg − Wm gk2 ≤ c (kg − Pm gk + kg − Pm
gk) kgk → 0, m → ∞,
as required.
We are now ready to analyze fn,m . First, note that we may rewrite (2.3) as
hfn,m , Wm φiR = hf, Wm φiR ,
∀φ ∈ Tn .
In other words, fn,m (whenever it exists) is precisely the oblique projection of f onto Tn along the subspace
[Wm (Tn )]⊥ (here ⊥ is taken with respect to (HR , h·, ·iR )). Letting
Cn,m = cos θTn ,Wm (Tn ) ,
(2.13)
we immediately deduce the following:
Theorem 2.11. Suppose that n, m ∈ N are such that Cn,m > 0, where Cn,m is given by (2.13). Then fn,m ,
as defined by (2.3), exists uniquely and satisfies the sharp bounds
kfn,m kR ≤
and
kf − fn,m kR ≤
1
Cn,m
1
Cn,m
kf kR ,
kf − Qn f kR .
(2.14)
(2.15)
Naturally, to use this theorem, we need to understand the quantity Cn,m . We have
Lemma 2.12. The quantity Cn,m defined by (2.13) satisfies
1
(1 − En,m ) ≤ Cn,m ≤ 1,
c
∀n, m ∈ N,
(2.16)
where c = supm∈N kPm k and k·k is the operator norm on the space of bounded linear maps (HR , h·, ·iR ) →
(HR , h·, ·iR ). Moreover, Cn,m → 1 as m → ∞ for fixed n.
Proof. That Cn,m ≤ 1 follows immediately from the definition (2.13). Moreover, by the definition of
subspace angles
Cn,m = inf kQWm (Tn ) φkR .
φ∈Tn
kφkR =1
Consider the quantity kQWm (Tn ) φkR . By the standard duality pairing,
hQWm (Tn ) φ, Wm φ0 iR
0
0
: φ ∈ Tn , φ 6= 0
kQWm (Tn ) φkR = sup
kWm φ0 kR
hφ, Wm φ0 iR
0
0
= sup
: φ ∈ Tn , φ 6= 0 .
kWm φ0 kR
Setting φ0 = φ, we obtain
kQWm (Tn ) φkR ≥
hφ, Wm φiR
kPm φk2R
=
.
kWm φkR
kWm φkR
10
(2.17)
∗
Since Wm = Pm
◦ Pm we obtain
kQWm (Tn ) φkR ≥
1
1
kPm φkR ≥ (1 − En,m ) kφk,
c
c
which gives (2.16). The proof that Cn,m → 1 as m → ∞ is analogous to the proof of Lemma 2.4, and
follows directly from the fact that Wm → I (Lemma 2.10).
The importance of (2.17) is that it allows Cn,m to be estimated in terms of En,m ; a somewhat easier task.
Thus, estimates obtained later for the asymptotic behaviour of En,m automatically hold for Cn,m , and thus
guarantee Theorem 2.11. We shall also exploit this fact in §3. Note that this lemma also implies asymptotic
optimality of fn,m : since
kfn,m − Qn f k2 = kf − fn,m k2 − kfn,m − Qn f k2 ≤ Kn,m kf − Qn f k2 ≤ Kn,m kf k2 ,
where Kn,m = C 21 − 1, we have fn,m → Qn f as m → ∞.
n,m
It is instructive to relate this general framework to the example of §2.1. Suppose that we sample an
analytic, nonperiodic f via its Fourier coefficients (so that Pm is the standard Fourier projection operator)
and wish to reconstruct in Chebyshev polynomials. We thus let V = L1 (−1, 1), HS = L2 (−1, 1) and
1
HR = L2w (−1, 1), where w(x) = (1 − x2 )− 2 is the Chebyshev weight on [−1, 1]. It transpires that both
(A1) and (A20 ) hold in this case (see §3). Therefore Theorem 2.11 and Lemma 2.12 apply to this example.
∗
Note that in this setting the operators Pm
and Wm are given explicitly by
∗
Pm
g=
2.3
1
Pm (wg),
w
Wm g =
1
Pm (wPm g) ,
w
g ∈ L2w (−1, 1).
(2.18)
Numerical implementation
Let us now consider the computation of fn,m . Writing fn,m =
easily seen that the vector α = (α1 , . . . , αn ) is defined by
Pn
j=1
αj φj for unknowns α1 , . . . , αn , it is
U ∗ CU α = U ∗ C fˆ,
where fˆ = {hf, ψ1 iS , . . . , hf, ψm iS }, and U ∈ Cm×n and C ∈ Cm×m have (j, k)th entries hφk , ψj iS and
hψj , ψk iR respectively. Thus the coefficients of fn,m are defined by an n × n linear system of equations,
with self-adjoint matrix A = U ∗ CU .
The condition number κ(A) of the matrix A is critical from a numerical perspective. It determines
both the stability of the numerical method – its susceptibility to noise and round-off errors, in particular
– as well as the cost of any iterative solver (conjugate gradients, for example) for computing the
pvector
of unknowns α. Specifically, the number of iterations required to compute α is proportional to κ(A).
Thus, if κ(A) = O (1) for all n and sufficiently large m (which turns out to be the case), then the cost of
computing fn,m is determined solely by the cost of performing matrix-vector multiplications involving A.
Since the (typically dense) matrices U and C are of size m × n and m × m respectively,
straightforward
implementation of this method, that is, without any fast transforms, requires O m2 operations. However, as
we shall explain in §4, C always has a Toeplitz structure when one considers Fourier samples. Thus, matrixvector multiplications involving C require only O (m log m) operations, giving a total figure of O (mn) for
this framework.
We now give an estimate for κ(A):
Lemma 2.13. Let En,m be given by (2.6). Then the condition number of A = U ∗ CU satisfies
κ(A) ≤
1 + En,m
1 − En,m
2
.
In particular, if n is fixed then A → I as m → ∞, where I ∈ Cn×n is the identity.
Proof.
α = {α1 , . . . , αn } ∈ Cn be an normalized eigenvector of A with corresponding eigenvalue λ. If
PLet
n
φ = j=1 αj φj it follows that
hPm φ, Pm φj iR = λαj ,
11
j = 1, . . . , n,
and, in particular, kPm φk2R = λ. Hence,
n
o
2
λmin (A) = inf kPm φkR : φ ∈ HR , kφkR = 1 ,
n
o
2
λmax (A) = sup kPm φkR : φ ∈ HR , kφkR = 1 .
Using the definition En,m we deduce that
(1 − En,m )2 ≤ λmin (A) ≤ λmax (A) ≤ (1 + En,m )2 ,
which gives the first result. For the second, we merely note that A → I componentwise as m → ∞ (since
En,m → 0 by Lemma 2.4), and therefore in norm.
This lemma confirms good conditioning of A whenever m is sufficiently large in comparison to n. In
fact, the condition number of A can be made arbitrarily close to unity by increasing m sufficiently.
Remark 2.14 As mentioned in §1.4, the extended GS framework introduced in this section can actually be
reinterpreted as a preconditioning of the original GS framework of [7]. In the original framework, one solves
U ∗ U α = U ∗ fˆ. However, unless (HR , h·, ·iR ) = (HS , h·, ·iS ) the matrix U ∗ U is not well-conditioned, leading to a higher cost in computing α. The extended framework merely replaces this with the preconditioned
equations U ∗ CU α = U ∗ C fˆ, where C is as above. This judicious choice of C preconditions the original
equations (by exploiting the orthogonality of both the sampling and reconstruction vectors) and therefore
gives the aforementioned improvement.
2.4
Scaling of m with n: the stable sampling rate
Observe that the same quantity En,m which determines the magnitude of the error committed by fn,m also
occurs in the estimate for the condition number. Indeed, provided m is chosen so that 1 − En,m is bounded
away from zero, we can ensure both quasi-optimality of fn,m and numerical stability. This motivates the
following definition. We let
Θ(n; θ) = min {m ∈ N : En,m < 1 − θ} ,
θ ∈ (0, 1).
For a given n, choosing m ≥ Θ(n; θ) ensures that
kf − fn,m kR ≤ 1 + θ−2 kf − Qn f kR + c(2 − θ)θ−2 |||f − Qn f |||,
and that
(2.19)
(2.20)
2
κ(A) ≤ 2θ−1 − 1 .
In other words, fn,m is quasi-optimal to f from Tn , and its computation is numerically stable, uniformly in
n ∈ N.
The quantity Θ(n; θ), originally introduced in [7] and named the stable sampling rate in [6], measures
how large m (the number of samples) must be for a given n for stable, quasi-optimal reconstruction. Importantly, Θ(n; θ) is computable. Indeed,
p
Lemma 2.15. Let En,m be given by (2.6). Then En,m = kBk, where B ∈ Cn×n is the Hermitian matrix
with (j, k)th entry hφj − Pm φj , φk − Pm φk iR .
Pn
Proof. For φ ∈ Tn , write φ = j=1 αj φj . Then
( Pn
2
En,m
)
αj αk hφj − Pm φj , φk − Pm φk iR
Pn
= sup
: α ∈ Cn , α 6= 0
j,k=1 αj αk hφj , φk iR
∗
α Bα
n
= sup
:
α
∈
C
,
α
=
6
0
.
α∗ α
j,k=1
This is precisely kBk.
Note that the matrix B can be expressed as B = I − V ∗ U − U ∗ V + A, where A = U ∗ CU and
V ∈ Cm×n has (j, k)th entry hφk , ψj iR . Hence, given U , C and V , one can readily compute En,m and
therefore Θ(n; θ).
12
The estimate (2.20) for kf − fn,m k holds with the stronger assumption (A20 ). In the case that this does
hold, then it is preferable to replace (2.20) by the sharp bound (2.15). Using (2.16) we obtain the improved
estimate
c
kf − fn,m kR ≤ kf − Qn f kR .
(2.21)
θ
In fact, one could, in theory, also use (2.15) directly, by numerically computing Cn,m (recall that Cn,m is
the angle between two finite-dimensional subspaces, and hence computable), thereby giving a sharp (and
consequently slightly better) bound.
The significance of Θ(n; θ) is that it allows one to control a priori both the stability and accuracy of the
numerical method. Whilst it is convenient that Θ(n; θ) can always be computed numerically, it is also important to have analytical estimates for each particular application. Naturally, such estimates are completely
dependent on both the sampling and reconstruction spaces, and thus must be obtained on a case-by-case
basis. We shall devote much of the second half of this paper to the derivation of such estimates for the case
of Fourier sampling and (piecewise) polynomial reconstructions.
3
Recovery from Fourier coefficients
We now consider the specific application of recovering a piecewise analytic function from its Fourier coefficients using a piecewise polynomial basis. As we show, it is possible to reconstruct in a completely stable
manner in an arbitrary basis of piecewise polynomials, orthogonal with respect to the (piecewise) modified
Jacobi weight. Thus, the individual polynomials employed can be specified by the user, making the method
extremely flexible in this sense. In §5 we consider a number of other related reconstruction problems which
can be addressed with this framework.
3.1
Preliminaries
Recall the setup of §2.1. Suppose that f : [−1, 1] → R is piecewise analytic with jumps at the known
locations −1 < x1 < . . . < xl < 1. Set x0 = −1 and xl+1 = 1. We shall assume that the first m Fourier
coefficients (2.4) of f are given. Thus, we let HS = L2 (−1, 1) with its standard inner product h·, ·i and norm
k·k, and define ψj (x) = √12 exp(ijπx) (recall that in this case we enumerate {ψj } over Z as opposed to N).
We shall reconstruct in the space Tn defined by (2.5), with a basis of this space consisting piecewise
polynomials orthogonal with respect to the piecewise (modified) Jacobi weight (1.3) with parameters α =
{α0 , . . . , αl }, β = {β0 , . . . , βl }, where αr , βr > −1, r = 0, . . . , l. We let HR = L2α,β (−1, 1) with
corresponding inner product h·, ·iα,β and norm k·kα,β .
We enumerate the orthonormal basis for Tn as follows. First, let cr = 21 (xr+1 − xr ) and define Λr (x) =
x−xr
cr − 1, so that Λ(Ir ) = [−1, 1], where Ir = [xr , xr+1 ]. For given αr , βr , let φj be the corresponding
orthonormal modified Jacobi polynomial of degree j on [−1, 1]. Assume that φj ≡ 0 outside [−1, 1] and
define the local (modified) Jacobi polynomial on Ir by φr,j = √1cr φj ◦ Λr . It follows that the set
{φr,j : j = 0, . . . , nr − 1, r = 0, . . . , l} ,
(3.1)
forms an orthonormal basis for Tn , and hence we may write fn,m as
fn,m =
l nX
r −1
X
αr,j φr,j ,
r=0 j=0
with unknowns αr,j ∈ C. We now compute fn,m via (2.3).
We first wish to analyze fn,m (implementation will be discussed in §4). This means applying the analysis
of §2, and for this we need to verify (A1) and (A2). Additionally, we also wish to determine when we may
replace (A2) by the stronger condition (A20 ), and hence when the sharp bounds of Theorem 2.11 apply.
Verifying these assumptions requires understanding the convergence of Fourier series in the weighted spaces
L2α,β (−1, 1). This is the content of the next section.
13
3.2
Convergence of Fourier series in weighted norms
Somewhat surprisingly, this question has been considered before. Convergence of Fourier series with respect
to piecewise modified Jacobi weights are simple consequences of this more general theory, which we now
recap. The reader is referred to [40, 43] or more recently [13], for further details.
Suppose that w(x) is a weight function on [−1, 1] with w(x) ≥ 0 almost everywhere and such that
0 < w(x) < ∞ almost everywhere in [−1, 1]. We shall denote the space of weighted square integrable
functions by L2w (−1, 1), with corresponding norm k·kw . We say that w(x) is a Helson–Szegö weight function
if and only if there exists a constant c > 0 such that
Z Z
1
1
1
w
≤ c,
|I| I
|I| I w
for all subsets I ⊆ [−1, 1] that are either subintervals or complements of subintervals in [−1, 1] (here |I|
denotes the length of I).
There is a fundamental relationship between Helson–Szegö weight functions and Fourier series. This is
summarized in the following theorem:
Theorem 3.1 (see [43] or [13]). The following four statements are equivalent:
(i) w is a Helson–Szegö weight function,
(ii) There exist real functions u, v on [−1, 1] with kuk∞ , kvk∞ < ∞ such that w(x) = eu(x)+ṽ(x) almost
everywhere,
(iii) There exists a finite constant c > 0 such that, for all f ∈ L2w (−1, 1), we have f ∈ L1 (−1, 1) and
kPm f kw ≤ ckf kw , ∀m ∈ N, where Pm f is the truncated Fourier expansion of f ,
(iv) For every f ∈ L2w (−1, 1), we have f ∈ L1 (−1, 1) and kf − Pm f kw → 0 as m → ∞.
Using this theorem (specifically conditions (iii) and (iv)), we immediately obtain the following:
Corollary 3.2. Let Pm be the Fourier projection operator. Then (A1) and (A20 ) hold for HR = L2w (−1, 1)
if and only if the weight function w is of Helson–Szegö type.
∗
◦ Pm , and Lemma 2.10 gives that
Remark 3.3 Recall that in §2.2.2 we introduce the operator Wm = Pm
R
Wm → I strongly on H . This can actually be proved quite easily in the case considered in this section. If
w is a Helson–Szegö weight function then, recalling the explicit form (2.18) for Wm ,
kg − Wm gkw ≤ kwg − Pm (wg)kw−1 + kPm (wg − wPm g)kw−1 .
1
If w is a Helson–Szegö weight function then so is w−1 (x) = w(x)
(wherever defined). Also, g ∈ L2 (−1, 1)
if and only if wg ∈ L2w−1 (−1, 1). Therefore Theorem 3.1 gives
kg − Wm gkw ≤ kwg − Pm (wg)kw−1 + ckg − Pm gkw → 0,
m → ∞,
as required.
We are interested in the case where (piecewise) modified Jacobi weight functions are of Helson–Szegö
type. We have
Lemma 3.4. Suppose that wα,β is a piecewise modified Jacobi weight function with parameters α =
{α0 , . . . , αl } and β = {β0 , . . . , βl }. Then wα,β is a Helson–Szegö weight function if and only if −1 <
αr , βr < 1 for all r = 0, . . . , l.
Proof. Note that if w̃ is a Helson–Szegö weight function and w w̃ uniformly in x, then so is w. We now
recall a result proved in [13]. Let w(t) = ϕ(eiπt ), where ϕ is finite, positive and continuous on the unit circle
except at a finite set of points z1 , . . . , zl . Moreover, suppose that
ϕ(z) = O (|z − zr |γr ) ,
1
= O |z − zr |−γr ,
ϕ(z)
z → zr .
Then w is a Helson–Szegö weight function if and only if −1 < γr < 1, r = 1, . . . , l. Noting that 1 − t2 |1 + eiπt | now gives the full result.
14
This lemma, in combination with Theorem 3.1, gives the following:
Corollary 3.5. Let Pm be the Fourier projection operator and HR = L2α,β (−1, 1). Then assumptions (A1)
and (A20 ) hold if and only if −1 < αr , βr < 1 for all r = 0, . . . , l.
Note that this corollary includes the cases α = β = {− 21 , . . . , − 12 , }, { 12 , . . . , 12 } corresponding to
Chebyshev weights (of the first and second kinds) that form the principal interest of this paper. With this
corollary in hand, the analysis of §2.2.2 immediately applies to this problem. In particular, we deduce
stability, convergence, asymptotic optimality and the sharp bounds of Theorem 2.11.
Let us now address the other case, where at least one αj or βj exceeds 1. As shown by Lemma 3.4, wα,β
is no longer a Helson–Szegö weight function. However, one can still verify (A1) and (A2) in this case, for a
suitable choice of the subspace U:
Corollary 3.6. Let Pm be the Fourier projection operator and wα,β a piecewise Jacobi weight function with
parameters α = {α0 , . . . , αl } and β = {β0 , . . . , βl }. For r = 0, . . . , l define
αr
−1 < αr < 1
βr
−1 < βr < 1
α̃r =
,
β̃r =
,
1 − r
αr ≥ 1
1 − 0r
βr ≥ 1
where 0 < r , 0r < 2 are arbitrary. Then (A1) and (A2) hold with U = L2α̃,β̃ (−1, 1) and ||| · ||| = k·kα̃,β̃ .
Proof. By construction, wα̃,β̃ is a piecewise modified Jacobi weight function of Helson–Szegö type. Moreover, we have the continuous embedding U ,→ HR . In other words, there exists c > 0 such that kgkα,β ≤
ckgkα̃,β̃ , ∀g ∈ U. Therefore, by Theorem 3.1,
kg − Pm gkα,β ≤ kg − Pm gkα̃,β̃ → 0,
∀g ∈ U,
and
kPm gkα,β ≤ kPm gkα̃,β̃ ≤ ckgkα̃,β̃ ,
∀g ∈ U.
This gives the result.
As a result of this corollary, the analysis of §2.2.1 applies in this setting. Hence one verifies stability and
convergence, albeit without the sharp bounds of §2.2.2. As commented, the space U of functions we can
reconstruct with this approach is very large, and certainly includes all functions of interest in practice.
Observe that Corollaries 3.5 and 3.6, in combination with the analysis of §2, confirm the first part of
the key theorem of this paper: namely, Theorem 1.4. For the second part, we need to estimate the stable
sampling rate Θ(n; θ), as defined by (2.19). This is the content of the next section. The final part of Theorem
1.4, i.e. the computational cost estimates, will be addressed in §4.
3.3
Estimates for the stable sampling rate Θ(n; θ)
The key theorem we prove in this section is the following:
Theorem 3.7. Let Pm be the Fourier projection operator, Tn be given by (2.5) for n = (n0 , . . . , nl ), and
suppose that HR = L2α,β (−1, 1), where α = {α0 , . . . , αl }, β = {β0 , . . . , βl }. Then, for 0 < θ < 1 there
exists c > 0 depending only on θ such that
Θ(n; θ) ≤ cn̄r ,
∀n0 , . . . , nl ∈ N,
where Θ(n; θ) is given by (2.19), n̄ = max{n0 , . . . , nl }, and
p−q
r = max 2, p + 1, 2 + 2
,
q+1
p = max{α0 , . . . , αl , β0 , . . . , βl },
q = min{α0 , . . . , αl , β0 , . . . , βl }.
A proof of this theorem is given later in this section. Recall that the most important case in this paper is
when α0 = . . . = αl = β0 = . . . = βl = γ. In other words, a piecewise ultraspherical weight with the same
parameter values in each of the intervals I0 , . . . , Il . In this case, p = q = γ, and Theorem 3.7 gives that
Θ(n; θ) ≤ cn̄max{2,γ+1} .
15
80
150
150
60
100
100
40
50
50
20
5
10
15
20
5
10
15
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
5
10
γ=
− 12
15
20
5
10
15
20
5
10
15
20
20
0.1
5
10
γ=
15
20
1
2
γ=2
Figure 1: The functions Θ(n; θ) (top) and n−2 Θ(n; θ) (bottom) against n = 1, . . . , 20 for θ =
(squares, circles and crosses respectively).
1 1 1
4 , 8 , 16
For example, whenever −1 < γ < 1, such as in the Legendre or Chebyshev cases, we obtain an O n2
scaling for Θ(n; θ). Note that this result was first proved in [42] (see also [7]) for the Legendre case.
In fact, quadratic scaling of Θ(n; θ) with n is the best possible predicted by this theorem (indeed, one
cannot obtain a better scaling whilst retaining stability with any method – see Remark 1.6 and [6]). One
negative consequence of this result is that if one chooses α and β so that either p 6= q or p > 1, then the
growth of Θ(n; θ) is more severe. Thus, more Fourier coefficients are required to reconstruct in terms of the
corresponding piecewise polynomial basis. This is clearly less than ideal. Nevertheless, whenever α and β
are chosen so that p = q and p ≤ 1, we retain quadratic growth.
The reason for the increased scaling whenever p > 1 is that the weighted norm k·kα,β is too weak in
comparison to the standard L2 norm to adequately control the error of the Fourier projection. Conversely,
when p 6= q, the local convergence of the Fourier series of an arbitrary function φ at any point x ∈ (−1, 1) is
determined by its global behaviour at every jump x0 , x1 , . . . , xl . Thus, convergence in the local norm over
each Ir subinterval must balance this global behaviour, which leads to the higher scaling when p 6= q.
In Figure 1 we plot Θ(n; θ) and n−2 Θ(n; θ) for case where l = 0 (i.e. analytic and nonperiodic functions
f ) and α = β = γ (i.e. reconstructions in ultraspherical polynomials). As is evident, when γ = ± 21 , Θ(n; θ)
scales quadratically with n. Conversely, this scaling is cubic when the value γ = 2 is used. Thus, these
results confirm Theorem 3.7.
The remainder of this section is devoted to a proof of Theorem 3.7. For this, we first require some general
results about Fourier series, and in particular, Fourier series of arbitrary piecewise algebraic polynomials.
For this, we require some additional notation. We shall write A . B to mean that there exists a constant
c > 0 independent of all relevant parameters such that A ≤ cB. Second, if I ⊆ [−1, 1] is an interval and
α, β > −1, we write k·k∞,I and k·kα,β,I for the norms corresponding to the spaces L2∞ (I) and L2wα,β (I)
respectively, where wα,β is a modified Jacobi weight, appropriately scaled to I. Finally, we let [g](x) =
g(x+ ) − g(x− ) be value of the jump in a function g : [−1, 1] → R at the point x ∈ [−1, 1], with the
understanding that [g](−1) = [g](1) = g(−1) − g(1).
Our first lemma is as follows:
Lemma 3.8. Suppose that f : [−1, 1] → R is piecewise smooth with singularities at −1 < x1 < . . . <
xl < 1. Then we have the asymptotic expansion
l
∞
1 X X (−1)s+1 e−iπjxr h (s) i
fˆj ∼ √
f
(xr ),
2 r=0 s=0 (−ijπ)s+1
16
|j| 1,
(3.2)
In particular, if f ∈ Tn , where Tn is as in (2.5), then
l nr −1
1 X X
(−1)s+1 e−iπjxr h (s) i
f
(xr ),
fˆj = √
(−ijπ)s+1
2 r=0 s=0
∀j ∈ Z\{0}.
Proof. This follows from repeated integration by parts applied to the integral
R1
−1
(3.3)
f (x)e−ijπx dx.
Note that the symbol ∼ in (3.2) denotes an asymptotic expansion (in the usual Poincaré sense) in the
parameter j → ±∞. As is typical, the right-hand side of (3.2) does usually not converge for any fixed j. On
the other hand, (3.3) holds for any finite j 6= 0 and arbitrary piecewise polynomials f .
A convenient means to describe the Fourier series of an arbitrary function piecewise smooth function
f is in terms of certain cardinal functions [3, 49]. For r ∈ N, let C̃r (x) be the periodic extension of
2r
Br+1 (x) to the real line, where Br+1 is the (r + 1)th Bernoulli polynomial [1, chpt. 23], and define
− (r+1)!
Cr (x) = C̃r ( 12 x + 1). It follows that Cr is a piecewise polynomial of degree r + 1 with singularity at x = 0.
Moreover, since the Bernoulli polynomials form an Appell sequence, we have that
i
h
Cr(s) (0) = δr−s , r, s ∈ N.
Define the piecewise polynomials
qr,s (x) = Cs (x − xr ),
r = 0, . . . , l,
s ∈ N,
via translation. Substituting f = qr,s in (3.3), we note that
1 (−1)s+1 e−ijπxr
,
qd
r,s j = √
2 (−ijπ)s+1
j ∈ Z\{0}.
(3.4)
Thus (3.2) can be expressed more succinctly as
fˆj ∼
∞
l X
X
h
i
(r)
qd
(xr ).
r,s j f
r=0 s=0
Note also that if φ ∈ Tn , where Tn is defined by (2.5), then
φ(x) − Pm φ(x) =
l nX
r −1 h
i
X
φ(s) (xr ) {qr,s (x) − Pm qr,s (x)} ,
(3.5)
r=0 s=0
Thus, to study the convergence of the Fourier series of an arbitrary φ ∈ Tn , it suffices to consider the Fourier
series of the cardinal functions qr,s . We use this approach when proving the next two results.
Lemma 3.9. Suppose that φ ∈ Tn . Then, there exists c > 0 independent of n and φ such that kφ −
Pm φk∞ . kφk∞ whenever m ≥ cn̄2 .
Proof. By (3.5) and Markov’s inequality [14],
kφ0 k . n2 kφk,
∀φ ∈ Pn−1 ,
(3.6)
we have
kφ − Pm φk∞ ≤
l nX
r −1 h
i
X
(s)
(xr ) kqr,s − Pm qr,s k∞
φ
r=0 s=0
l nX
r −1
X
2s
.
nr kφk∞,Ir + n2s
r−1 kφk∞,Ir−1 kqr,s − Pm qr,s k∞ ,
(3.7)
r=0 s=0
where, for convenience we let I−1 = Il and n−1 = nl . Consider qr,s . For s = 1, 2, . . . , the function
00
qr,s ∈ C1 (T) and the derivative qr,s
is of bounded variation. Hence Pm qr,s converges uniformly to qr,s , the
coefficients qd
r,s j are absolutely summable and we may write qr,s (x) − Pm qr,s (x) as the infinite sum
X
qr,s (x) − Pm qr,s (x) =
qd
∀x ∈ [−1, 1].
r,s j ψj (x),
|j|≥ m
2
17
Using the expression (3.4) we deduce that
X
kqr,s − Pm qr,s k∞ .
|d
qr,s j | .
X
j −s−1 . m−s .
j≥ m
2
|j|≥ m
2
Now consider qr,0 . By translation, it suffices to consider q0,0 (x) = 14 − 12 x. This function clearly has
uniformly bounded Fourier series. Hence, substituting this and the previous result into (3.7), we obtain
kφ − Pm φk∞ . kφk∞
s
l nX
r −1 X
c̃n2
r
m
r=0 s=0
,
for some c̃ > 0. Therefore, provided m ≥ cn̄2 with c > c̃, this sum is bounded independently of n, m and
the result follows.
Lemma 3.10. Suppose that φ ∈ Tn and let x ∈ (xr , xr+1 ) for some r = 0, . . . , l. Then, there exists c > 0
independent of n, r, φ and x such that
|φ(x) − Pm φ(x)| . m−1 kφk∞
l
X
1 + | sin π2 (x − xr )|−1 ,
r=0
whenever m ≥ cn̄2 .
Proof. Suppose that we can show that
|qr,s (x) − Pm qr,s (x)| . m−s−1 1 + | sin π2 (x − xr )|−1 ,
s ∈ N, r = 0, . . . , l,
x 6= xr .
(3.8)
Then the result follows directly from (3.5) and Markov’s inequality (3.6):
|φ(x) − Pm φ(x)| .
l
X
1 + | sin
π
2 (x
− xr )|
r=0
−1
r −1
nX
2s
m−s−1 n2s
r + nr−1 kφk∞
s=0
. m−1 kφk∞
l
X
1 + | sin π2 (x − xr )|−1
r=0
r −1
nX
s=0
n̄2
m
s
.
Hence it remains to show (3.8). Without loss of generality, we consider q0,s (x). Using (3.4) and noticing
that Pm q0,s (x) converges to q0,s (x) whenever −1 < x < 1, we obtain
X zj 1 X (−1)j
,
|q0,s (x) − Pm q0,s (x)| =
exp(ijπx) . s+1 2π s+1 j s+1
j
j>m
|j|>m
where z = − exp(iπx) (replacing b m
2 c by m in the summation does not effect results and slightly simplifies
the notation). Thus, it suffices to consider the sum
Ss,m (z) =
X zj
.
j s+1
j>m
We now use Abel summation:
X
X zj 1
X
X
1
zj
zj
zj
−
+
=
+
.
Ss,m (z) =
js j
j+1
j s (j + 1) j>m j s+1 (j + 1) j>m j s (j + 1)
j>m
j>m
The first term is O m−s−1 uniformly in |z| = 1. Now consider the second sum. We have
z −1
X zj
X zj
zj
= z −1
1 + O j −1 = z −1
+ O m−s−1 ,
s
s+1
s+1
j(j − 1)
j
j
j>m+1
j>m+1
j>m+1
X
18
which holds uniformly in |z| = 1. Hence
zm
Ss,m (z) = z −1 Ss,m (z) − s+1 + O m−s−1 .
m
Rearranging, this gives
zm
m−s−1 + O m−s−1 .
1−z
Note that x 6= ±1, therefore z =
6 1. Moreover, since z = − exp(iπx), we have that
Ss,m (z) =
|z − 1| = 2| cos π2 x| = 2| sin π2 (x − x0 )|,
(recall that x0 = −1) and consequently
|Ss,m (z)| . 1 + | sin π2 (x − x0 )|−1 m−s−1 ,
which gives (3.8).
We are now able to prove the main result of this section:
Proof of Theorem 3.7. It suffices to show that, given θ ∈ (0, 1), there exists c such that En,m < 1 − θ for all
m ≥ cn̄r , where En,m is given by (2.6). Let φ ∈ Tn with kφkR = 1. Note that
kφ − Pm φk2α,β =
l Z
X
r=0
|φ(x) − Pm φ(x)|2 wαr ,βr (x) dx.
Ir
We consider each term of this sum separately. Write Ir = Ir+ ∪ Ir− , where Ir+ = (dr , xr+1 ), Ir− = (xr , dr )
and dr = 21 (xr + xr+1 ) is the midpoint of Ir . In a similar manner we also define, for arbitrary 0 < < cr =
1
2 (xr+1 − xr ),
−
Ir,
= (xr + , dr ),
+
Ir,
= (dr , xr+1 − ),
+
Jr,
= (xr+1 − , xr+1 ),
−
Jr,
= (xr , xr + ).
We now write
Z
|φ(x)−Pm φ(x)|2 wαr ,βr (x) dx
Ir
Z
Z
2
|φ(x) − Pm φ(x)| wαr ,βr (x) dx +
=
−
Ir,
+
Ir,
Z
+
+
Jr,
|φ(x) − Pm φ(x)|2 wαr ,βr (x) dx +
|φ(x) − Pm φ(x)|2 wαr ,βr (x) dx
Z
−
Jr,
|φ(x) − Pm φ(x)|2 wαr ,βr (x) dx.
+
Consider the integral over Ir,
. By construction wα,β (x) . (xr+1 − x)αr uniformly for x ∈ Ir+ . Using this
and Lemma 3.10 we find that
Z
l Z
X
|φ(x) − Pm φ(x)|2 wαr ,βr (x) dx . m−2 kφk2∞
1 + | sin π2 (x − xs )|−1 (xr+1 − x)αr dx.
+
Ir,
s=0
+
Ir,
Since | sin π2 (x − xr+1 )| & |x − xr+1 | and | sin π2 (x − xs )| & 1, s 6= r + 1, whenever −1 < |x − xr+1 | < 1
we deduce that
!
Z
Z
2
−2
2
αr −2
|φ(x) − Pm φ(x)| wαr ,βr (x) dx . m kφk∞ 1 +
(xr+1 − x)
dx
+
Ir,
+
Ir,
. max{1, αr −1 }m−2 kφk2∞ .
−
Similarly, for Ir,
, we have
Z
|φ(x) − Pm φ(x)|2 wαr ,βr (x) dx . max{1, βr −1 }m−2 kφk2∞ .
−
Ir,
19
We now require an estimate for kφk∞ . First let p ∈ Pn be a polynomial and suppose that α, β > −1. Then,
it can be shown that
kpk∞ . nmax{α,β}+1 kpkα,β .
Pn
(this follows directly from the decomposition kpk2α,β = j=0 |hp, pj iα,β |2 , where {pj } are the orthonormal
1
polynomials with respect to the weight function wα,β , and the fact that kpj k∞ ∼ j max{α,β}+ 2 [48]).
Since φ ∈ Tn , and therefore φ|Ir ∈ Pnr −1 , we have
n
o
kφk∞ = max kφk∞,Ir . max nrmax{αr ,βr }+1 kφkαr ,βr ,Ir ≤ n̄p+1 kφkα,β ,
r=0,...,l
r=0,...,l
where p = max{α0 , . . . , αl , β0 , . . . , βl }. Hence,
(Z
Z
l
X
2
|φ(x) − Pm φ(x)| wαr ,βr (x) dx +
−
Ir,
+
Ir,
r=0
. max{1, p−1 }
n̄p+1
m
2
)
|φ(x) − Pm φ(x)|2 wαr ,βr (x) dx
kφk2α,β .
±
Now consider the integral over Jr,
. After an application of Lemma 3.9, we obtain
Z 1
Z
2
2
(1 − x)αr dx . kφk2∞ αr +1 ,
|φ(x) − Pm φ(x)| wαr ,βr (x) dx . kφk∞
+
Jr,
1−
−
and likewise for Jr,
with the exponent αr replaced by βr . Since < 1, this gives
)
(
Z
Z
l
X
2
2
|φ(x) − Pm φ(x)| wαr ,βr (x) dx . q+1 n̄2(p+1) kφk2α,β .
|φ(x) − Pm φ(x)| wαr ,βr (x) dx +
r=0
−
Jr,
+
Jr,
Setting = cm−1 for c < min{cr : r = 0, . . . , l} and combining terms, we obtain
"
#
p+1 2
2(p+1)
n̄
n̄
kφ − Pm φk2α,β . max{1, m1−p }
+
kφk2α,β .
m
mq+1
Hence
"
2
En,m
=
sup
φ∈Tn
kφkα,β =1
kφ −
Pm φk2α,β
1−p
. max{1, m
}
n̄p+1
m
2
#
n̄2(p+1)
+
.
mq+1
(3.9)
The second term can be made arbitrarily small by selecting
p+1
p−q
m & n̄2 q+1 = n̄2+2 q+1 .
Now consider the first term. If p < 1 then this is
p+1 2 2 p+1
n̄
n
=
,
m1−p
m
m
and therefore we require m & n̄2 . Conversely, when p ≥ 1, we have
p+1 2 p+1 2
n̄
n̄
1−p
max{1, m }
=
,
m
m
and we therefore stipulate that m & n̄p+1 . Hence, the first term of the right-hand side of (3.9) can be made
arbitrarily small by the choice m & n̄max{2,p+1} . This completes the proof.
One downside of Theorem 3.7 is it does not give an explicit upper bound for the scaling constant c. In
simpler case of Legendre polynomials explicit, and fairly sharp, bounds were derived in [7]. It is, in theory,
possible to repeat the proof of Theorem 3.7 to give an explicit bound. However, because the proof is rather
involved, this is likely to be grossly pessimistic. On the other hand, since Θ(n; θ) can always be computed,
we may always numerically estimate c. In fact, as shown in Figure 1 one can take a fairly small value c = 41
in implementations. We use this value in the next section.
20
4
4.1
Numerical implementation and examples
Implementation
As mentioned in §2.3, the main issue in implementation is the computation of the entries of the matrices U
and C. Computation of the entries of U can be carried out using the iterative procedure of [7, Lemma 3.4].
The total cost incurred is O (mn) operations. Note that, as commented in [7], one may in practice need to
use a two-phase algorithm (such as that described in [21]) to ensure stability of the computation of these
entries. However, this does not increase the overall cost.
For the computation of C, recall that this matrix has (j, k)th entry hψj , ψk iR . In other words, if wα,β is
the corresponding weight function, then
1
hψj , ψk iR =
2
Z
1
eiπ(j−k)x wα,β (x) dx = cj−k .
−1
Hence C is always a Toeplitz matrix, and to compute C we only need to determine cj for j = 0, . . . , m (note
that c−j = cj ). Fortunately, since
Z
1
eizx (1 − x2 )α dx =
−1
√ Γ(α + 1)
π
(α + 12 )
Γ(α + 21 )
α+ 12
2
Jα+ 12 (z),
z
z ∈ R,
where Γ is the Gamma function and Jν is the Bessel function of the first kind [35], we can always determine
cj explicitly for the case where wα,β is a piecewise ultraspherical weight function. Unfortunately, there is
no such closed form expression when wα,β is a piecewise Jacobi weight. However, in this case one could
always compute cj using aGaussian quadrature based on the given weight function, with the total cost of
this approach being O m2 . We shall not discuss this further.
For evaluation of the quantity Θ(n; θ) it is also necessary to compute the matrix V ∈ Cm×n with (j, k)th
entry hφk , ψj iR . Whenever w corresponds to a piecewise ultraspherical weight function, the entries of V are
actually known explicitly. This follows from the fact that the integrals
Z
1
eizx φj (x)(1 − x2 )α dx,
z ∈ R,
−1
have an explicit expression whenever φj is the j th normalized ultraspherical polynomial corresponding to
the parameter α > −1 (see, for example, [35, eqn. (3.7)]).
Note that the above arguments establish the final part of Theorem 1.4 (the first parts were addressed in
the previous sections) concerning the cost of the preprocessing step in the computation of fn,m (i.e. forming
the matrices U and C).
4.2
Examples
In Figure 2 we consider the reconstruction of the function f (x) = ex cos 8x using Legendre polynomials and
Chebyshev polynomials of the first and second kinds. As is evident, all three approaches give near identical
approximation errors. In fact, regardless of the polynomial basis used, one attains roughly machine accuracy
using only m = 225 Fourier samples (by means of comparison, direct expansion of this function in a Fourier
series gives an error of order 10−1 for this value of m). Also, the condition number of the matrix A = U ∗ CU
remains bounded in n, as predicted by the analysis in §2.
In Figure 3 we consider the piecewise analytic function
(2e2π(x+1) − 1 − eπ )(eπ − 1)−1 x ∈ [−1, − 21 )
f (x) =
(4.1)
π
x ∈ [− 12 , 1]
− sin( 2πx
3 + 3)
This function was put forth in [53] to test algorithms for overcoming the Gibbs phenomenon. Aside from the
discontinuity, its sharp peak makes it a challenging function to reconstruct accurately. However, as shown
in this figure, using a polynomial of degree 16 in each subinterval of smoothness, we recover the function
to around 14 digits of accuracy. As in the previous example, all three polynomial bases used (Legendre and
Chebyshev) give roughly the same approximation error, and as before, lead to bounded condition numbers
with κ(A) being at worst ≈ 10 for all cases.
21
20
1.0
5
10
15
20
25
30
35
0.001
0.8
15
0.6
10
10
-6
0.4
10-9
0.2
10-12
5
5
10
15
kf − fn,m k∞
20
25
30
35
5
10
15
En,m
20
25
30
35
κ(A)
Figure 2: The quantities kf − fn,m k∞ , En,m and κ(A) (left to right) against n = 1, 2, . . . , 40, for f (x) =
ex cos 8x, m = d 41 n2 e and γ = − 12 , 0, 12 (squares, circles and crosses respectively).
1.0
5
10
15
0.5
-0.5
1.0
-0.5
-1.0
f (x)
25
8
10-6
-1.0
20
10
0.001
0.5
6
10-9
4
10-12
2
10-15
5
kf − fn,m k∞
10
15
20
25
κ(A)
Figure 3: The function (4.1) (left), and the quantities kf − fn,m k∞ and κ(A) against n = 1, 2, . . . , 30, with
n0 = n1 , m = n20 and γ = − 21 , 0, 12 (squares, circles and crosses respectively).
This function was used in [30] to test spectral reprojection. Therein it was found that more than 1024
Fourier coefficients were required to obtain close to machine precision. Conversely, with the GS approach
we achieve the same value using only around 256 such coefficients – a factor of four increase (a similar
experiment was reported in [7]). Moreover, it is worth recalling that (see §1), with this approach one has
near-complete freedom to choose the polynomial basis for reconstruction, whereas with spectral reprojection
one must use a specific, Gibbs complementary, basis.
5
Other applications
As mentioned, the extended GS framework of §2 is far more general than the application considered in the
previous section of recovering a function from its Fourier coefficients. In this section we briefly discuss a
family of other related problem to which this framework can also be successfully applied.
Let f : [−1, 1] → R br piecewise analytic, and suppose now that we have access to the first m coefficients of f with respect to a basis of orthogonal polynomials (e.g. Jacobi or ultraspherical). Since f is only
piecewise analytic, direct expansion in the given basis of polynomials yields an extremely poor approximation (in fact, it suffers from a Gibbs-type phenomenon). Therefore, much as in the Fourier case, the problem
is to recover f to high accuracy (other variants of this problem have also been considered, such as the reconstruction of functions from Fourier–Bessel [47] and spherical harmonic coefficients [27]). We remark
that this problem occurs notably in spectral methods for hyperbolic PDEs [31, 34, 41]. Although spectral
reprojection can also be used in this setting [32, 33, 34], it suffers from the same drawbacks as those in the
Fourier case (see §1). Thus, we now propose to use GS instead.
Let {ψj } be orthonormal polynomials with respect to the (modified) Jacobi weight function wαS ,β S ,
where αS , β S > −1, and let {φj } be piecewise orthonormal polynomials with respect to the piecewise
(modified) Jacobi weight function wαR ,β R , where αR = {α0 , . . . , αl } and β R = {β0 , . . . , βl }. We shall
consider the following two important cases:
(i) αS = β S = γ S = 0,
(ii) αS = β S = γ S = − 21 or αS = β S = γ S = 21 ,
corresponding to sampling f with Legendre and Chebyshev polynomials (of the first and second kind) respectively.
22
5.1
Legendre polynomial sampling
As in the Fourier sampling case, to apply the GS framework of §2 we need to verify (A1) and (A2) for this
example. Note that HS = L2 (−1, 1) in this setting, and HR = L2γ R (−1, 1). We have:
Theorem 5.1. Suppose that HS = L2 (−1, 1) and Pm is the Legendre polynomial projection operator.
Let αR = {α0 , . . . , αl } and β R = {β0 , . . . , βl } with α0 , . . . , αl , β0 , . . . , βl > −1 arbitrary, and define
HR = L2αR ,β R (−1, 1). Let
U = f : f |Ir ∈ H1 (Ir ), r = 0, . . . , l ,
be the space of all piecewise H1 functions, and define ||| · ||| to be the standard norm on U. Then (A1) and
(A2) hold.
Proof. For α > −1 it can be shown that
Z
1
|g(x)|2 (1 ± x)α dx .
−1
kgk2
kgk2(1+α) kgk−2α
∞
α≥0
−1 < α < 0
(see [7, Lemma 3.6]), provided g ∈ L∞ (−1, 1). Now, for Ir = (xr , xr+1 ) define Ir− = (xr , dr ), Ir+ =
(dr , xr+1 ), where dr = 21 (xr + xr+1 ), so that Ir = Ir+ ∪ Ir− . Also, let
J1− = {r : −1 < βr < 0},
J1+ = {r : −1 < αr < 0},
J2+ = {r : αr ≥ 0},
J2− = {r : βr ≥ 0}.
Then, for g ∈ U,
kgkαR ,β R .
X
r
r
kgk1+α
kgk−α
+
0,I +
∞,I +
r
r
r∈J1+
X
1+βr
−βr
kgk0,I
+
− kgk
∞,I −
r
r∈J1−
X
r
kgk0,Ir+ +
r∈J2+
X
kgk0,Ir− ,
r∈J2−
where k·k0,I and k·k∞,I are the L2 and uniform norms on an interval I respectively. It follows that
kgkαR ,β R .
X
−αr
+
kgk1+αr kgk∞
X
r
kgk1+βr kgk−β
∞ + kgk
(5.1)
r∈J1−
r∈J1+
Now, for a function g ∈ U, it can be shown that its Legendre polynomial expansion, whilst exhibiting
a Gibbs-type phenomenon, is uniformly bounded and satisfies kPm gk∞ . |||g||| [46]. Now, suppose we
replace g by Pm g in (5.1). Then, using this bound and Bessel’s inequality kPm gk ≤ kgk ≤ |||g|||, we obtain
kgkαR ,β . |||g|||, which gives (A2). Similarly, replacing g by g − Pm g and using the observations that
kg − Pm gk → 0 as m → ∞ and that kg − Pm gk∞ ≤ kgk∞ + kPm gk∞ ≤ c|||g|||, we obtain (A1).
This theorem shows that, provided we restrict to a space of sufficiently regular functions (which, in practice, will always be the case), we can always recover a function f in any polynomial basis from its Legendre
polynomial samples. Note that the choice of U here is not unique, and the smoothness assumption could
most likely be lowered with a deeper understanding of the convergence of Legendre polynomial expansions
for piecewise functions. However, since we mainly consider piecewise analytic functions (which always
belong to this choice of U), this consideration is of little importance.
We shall omit two further considerations from this discussion. First, it is unknown at this moment under
what conditions the stronger assumption (A20 ) hold for this problem. Second, we also have no estimate for
the quantity Θ(n; θ). In [7] the case γ R ≡ 0 (that is, we both sample and reconstruct using
(piecewise)
Legendre polynomials) was considered. It was shown numerically that Θ(n; θ) = O n2 when γ R ≡ 0.
Establishing this result remains an open problem, as does determining Θ(n; θ) for other values of γR .
5.2
Chebyshev polynomial sampling
We now consider the case where γ S = ± 21 . In this case, unlike that of Legendre polynomial sampling, we
are actually able to provide a complete answer, at least regards assumptions (A1) and (A2) (or (A20 )). This
follows directly from the fact that the Chebyshev polynomial expansion of a function can be expressed as a
Fourier expansion under the transformation x = cos θ:
23
Theorem 5.2. Suppose that HS = L2γ (−1, 1) and Pm is the projection operator corresponding to Chebyshev
polynomials of the first (γ = − 12 ) or second (γ = 12 ) kind. Let w be an integrable weight function and
set HR = L2w (−1, 1). Then (A1) and (A20 ) hold if and only if the function w(cos θ) sin θ (first kind) or
w(cos θ)/ sin θ (second kind) is a Helson–Szegö weight function on [0, π].
Proof. Consider the case of first kind Chebyshev polynomials. The projection Pm f of a function f ∈
L2− 1 (−1, 1) is precisely the Fourier expansion the function F (θ) = f (cos θ) on [0, π]. Note that f ∈
2
L2− 1 (−1, 1) if and only if F ∈ L2 (0, π). Hence, the result follows immediately from Corollary 3.2. The
2
case where γ =
1
2
is similar.
Corollary 5.3. Suppose that HS = L2γ (−1, 1) and Pm is the projection operator corresponding to Chebyshev polynomials of the first (γ = − 12 ) or second (γ = 21 ) kind. Then (A1) and (A20 ) hold for the piecewise
(modified) Jacobi weight function wα,β if and only if
−1 < αr , βr+1 < 1,
r = 0, . . . , l − 1,
and −1 < αl , β0 < 0 (γ = − 21 ) or 0 < αl , β0 < 2 (γ = 12 ).
Proof. Let γ = − 12 . By Theorem 5.2 it suffices to consider the weight function W (θ) = wα,β (cos θ) sin θ
on [0, π]. Let θr = arccos(xr ), r = 1, . . . , l. Then
W (θ) = O (θr − θ)βr+1 , θ → θr− .
W (θ) = O ((θ − θr )αr ) , θ → θr+ ,
Also
W (θ) = O θ1+αl ,
θ → 0+ ,
W (θ) = O (π − θ)1+β0 ,
θ → π− .
Hence, as in the proof of Lemma 3.4, W is a Helson–Szegö weight function if and only if −1 < αr , βr+1 <
1, r = 0, . . . , l − 1 and −1 < αl , β0 < 0. The case of γ = 12 is similar.
Due to this theorem, it is always possible to reconstruct a function in a basis of piecewise Jacobi polynomials from its Chebyshev polynomial samples. In particular, we can recover in terms of piecewise Chebyshev
polynomials α0 = . . . = αl = β0 = . . . = βl = γ = ± 21 from Chebyshev polynomial samples. Note that,
unlike the Legendre polynomial case (see §5.1), we have (A20 ) for this problem, as opposed to the weaker
condition (A2).
6
Conclusions and challenges
In this paper we have introduced a general framework for sampling and reconstruction in Hilbert spaces,
and analyzed in detail the resulting numerical method for reconstructing a piecewise analytic function in a
orthogonal basis of (piecewise) polynomials from its Fourier coefficients. The method can be implemented
with a wide variety of different polynomial bases, including Chebyshev and Legendre polynomial bases. In
all cases the numerical method is stable and exponentially convergent (in the polynomial degree n). Whenever Chebyshev or Legendre polynomials are employed, for example, the method is also root-exponentially
convergent in the number of Fourier coefficients m. Finally, we have given a first insight into how the same
framework can be applied to the related problem of reconstructing a piecewise analytic function from its
orthogonal polynomial coefficients.
Purposefully we have not described the implementation of this framework for the polynomial coefficient
reconstruction problem described in §5. Nor have we presented numerical examples. The main stumbling
block herein is the computation of the entries of the matrices U and C. This involves constructing particular
quadratures to approximate the inner products hψj , φk iS and hψj , ψk iR , and is beyond the scope of this
paper. This aside, we have also not addressed the behaviour of the quantity Θ(n; θ) for this problem. In [7]
numerical results were presented suggesting quadratic growth in the case HR = HS = L2 (−1, 1) (i.e. Legendre polynomial sampling and reconstruction). This, and the corresponding extension to other orthogonal
polynomial systems, remains a conjecture.
This aside, it is worth repeating that the GS framework developed in this paper is far more general than
the problems considered in §3–§5. Namely, it allows one to reconstruct any element f ∈ U from its samples in an arbitrary orthonormal basis, even in the case that the sampling and reconstruction vectors are
24
orthogonal with respect to different inner products. In fact, this paper forms part of a much larger project
on numerical methods for sampling and reconstruction (see [4, 5, 6, 7, 8]), with a long list of potential applications. Currently we are investigating a number of such applications, including spline and wavelet-based
reconstructions of images, the solution of certain inverse and ill-posed problems (see [8]), and reconstructions from other types of integral transforms. It is also worth mentioning that the key ideas for GS originated
from new tools for the computational spectral problem [38, 39]. Furthermore, similar ideas, when combined
with convex optimization techniques, can be used to extended the current theory of compressed sensing to
formally infinite-dimensional problems. This topic is described in [5].
References
[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover, 1974.
[2] B. Adcock. Convergence acceleration of modified Fourier series in one or more dimensions. Math. Comp.,
80(273):225–261, 2011.
[3] B. Adcock. Gibbs phenomenon and its removal for a class of orthogonal expansions. BIT, 51(1):7–41, 2011.
[4] B. Adcock and A. C. Hansen. A generalized sampling theorem for stable reconstructions in arbitrary bases. J.
Fourier Anal. Appl. (to appear), 2010.
[5] B. Adcock and A. C. Hansen. Generalized sampling and infinite-dimensional compressed sensing. Technical report
NA2011/02, DAMTP, University of Cambridge, 2011.
[6] B. Adcock and A. C. Hansen. Sharp bounds, optimality and a geometric interpretation for generalised sampling in
Hilbert spaces. Technical report NA2011/10, DAMTP, University of Cambridge, 2011.
[7] B. Adcock and A. C. Hansen. Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon.
Appl. Comput. Harmon. Anal. (to appear), 2011.
[8] B. Adcock, A. C. Hansen, E. Herrholz, and G. Teschke. Generalized sampling: extension to frames and ill-posed
problems. in preparation, 2011.
[9] R. Archibald, K. Chen, A. Gelb, and R. Renaut. Improving tissue segmentation of human brain MRI through
preprocessing by the Gegenbauer reconstruction method. NeuroImage, 20(1):489–502, 2003.
[10] R. Archibald and A. Gelb. A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue
boundary integrity. IEEE Transactions on Medical Imaging, 21(4):305–319, 2002.
[11] D. Batenkov and Y. Yomdin. Algebraic Fourier reconstruction of piecewise smooth functions. Math. Comp. (to
appear), 2011.
[12] B. Beckermann, V. Kalyagin, A. Matos, and F. Wielonsky. How well does the Hermite–Padé approximation smooth
the Gibbs phenomenon? Math. Comp., 80:931–958, 2011.
[13] N. Borovykh and M. N. Spijker. Bounding partial sums of Fourier series in weighted L2 -norms, with applications
to matrix analysis. J. Comput. Appl. Math., 147(2):349–368, 2002.
[14] P. Borwein and T. Erdélyi. Polynomials and Polynomial Inequalities. Springer–Verlag, New York, 1995.
[15] J. P. Boyd. Chebyshev and Fourier Spectral Methods. Springer–Verlag, 1989.
[16] J. P. Boyd. Trouble with Gegenbauer reconstruction for defeating Gibbs phenomenon: Runge phenomenon in the
diagonal limit of Gegenbauer polynomial approximations. J. Comput. Phys., 204(1):253–264, 2005.
[17] J. P. Boyd. Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of
1/n. J. Comput. Phys., 228:1404–1411, 2009.
[18] C. Brezinski. Extrapolation algorithms for filtering series of functions, and treating the Gibbs phenomenon. Numer.
Algorithms, 36:309–329, 2004.
[19] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods: Fundamentals in Single Domains.
Springer, 2006.
[20] O. Christensen. An Introduction to Frames and Riesz Bases. Birkhauser, 2003.
[21] V. Dominguez, I. G. Graham, and V. P. Smyshlyaev. Stability and error estimates for Filon-Clenshaw-Curtis rules
for highly-oscillatory integrals. IMA J. Num. Anal. (to appear), 2011.
[22] T. A. Driscoll and B. Fornberg. A Padé-based algorithm for overcoming the Gibbs phenomenon. Numer. Algorithms, 26:77–92, 2001.
[23] K. S. Eckhoff. Accurate and efficient reconstruction of discontinuous functions from truncated series expansions.
Math. Comp., 61(204):745–763, 1993.
25
[24] K. S. Eckhoff. Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions.
Math. Comp., 64(210):671–690, 1995.
[25] K. S. Eckhoff. On a high order numerical method for functions with singularities. Math. Comp., 67(223):1063–
1087, 1998.
[26] J. B. Garnett. Bounded analytic functions. Springer, New York, 2007.
[27] A. Gelb. The resolution of the Gibbs phenomenon for spherical harmonics. Math. Comp., 66(218):699–717, 1997.
[28] A. Gelb and S. Gottlieb. The resolution of the Gibbs phenomenon for Fourier spectral methods. In A. Jerri, editor,
Advances in The Gibbs Phenomenon. Sampling Publishing, Potsdam, New York, 2007.
[29] A. Gelb and E. Tadmor. Detection of edges in spectral data. Appl. Comput. Harmon. Anal., 7(1):101, 1999.
[30] A. Gelb and J. Tanner. Robust reprojection methods for the resolution of the Gibbs phenomenon. Appl. Comput.
Harmon. Anal., 20:3–25, 2006.
[31] D. Gottlieb and J. S. Hesthaven. Spectral methods for hyperbolic problems. J. Comput. Appl. Math., 128(1-2):83–
131, 2001.
[32] D. Gottlieb and C.-W. Shu. On the Gibbs phenomenon IV: Recovering exponential accuracy in a subinterval from
a Gegenbauer partial sum of a piecewise analytic function. Math. Comp., 64(211):1081–1095, 1995.
[33] D. Gottlieb and C.-W. Shu. On the Gibbs phenomenon III: Recovering exponential accuracy in a sub- interval from
a spectral partial sum of a piecewise analytic function. SIAM J. Num. Anal., 33(1):280–290, 1996.
[34] D. Gottlieb and C.-W. Shu. On the Gibbs’ phenomenon and its resolution. SIAM Rev., 39(4):644–668, 1997.
[35] D. Gottlieb, C.-W. Shu, A. Solomonoff, and H. Vandeven. On the Gibbs phenomenon I: Recovering exponential
accuracy from the Fourier partial sum of a nonperiodic analytic function. J. Comput. Appl. Math., 43(1–2):91–98,
1992.
[36] B.-Y. Guo, J. Shen, and L.-L. Wang. Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J.
Sci. Comput., 27(1–3):305–322, 2006.
[37] B.-Y. Guo, J. Shen, and L.-L. Wang. Generalized Jacobi polynomials/functions and their applications. Appl. Numer.
Math., 59:1011–1028, 2009.
[38] A. C. Hansen. On the approximation of spectra of linear operators on Hilbert spaces. J. Funct. Anal., 254(8):2092–
2126, 2008.
[39] A. C. Hansen. On the solvability complexity index, the n-pseudospectrum and approximations of spectra of operators. J. Amer. Math. Soc., 24(1):81–124, 2011.
[40] H. Helson and Szegö. A problem in prediction theory. Ann. Mat. Pura Appl., 51(107–138), 1960.
[41] J. S. Hesthaven, S. Gottlieb, and D. Gottlieb. Spectral Methods for Time–Dependent Problems. Cambridge University Press, 2007.
[42] T. Hrycak and K. Gröchenig. Pseudospectral Fourier reconstruction with the modified inverse polynomial reconstruction method. J. Comput. Phys., 229(3):933–946, 2010.
[43] R. Hunt, B. Muckenhoupt, and R. Wheeden. Weighted norm inequalities for the conjugate function and Hilbert
transform. Trans. Amer. Math. Soc., 176(227–251), 1973.
[44] J.-H. Jung and B. D. Shizgal. Towards the resolution of the Gibbs phenomena. J. Comput. Appl. Math., 161(1):41–
65, 2003.
[45] J.-H. Jung and B. D. Shizgal. Generalization of the inverse polynomial reconstruction method in the resolution of
the Gibbs phenomenon. J. Comput. Appl. Math., 172(1):131–151, 2004.
[46] S. M. Kaber. The Gibbs phenomenon for Jacobi expansions. Commun. App. Anal., 10(2-3):133–148, 2006.
[47] J. Kamm, T. Williams, J. Brock, and S. Li. Application of Gegenbauer polynomial expansions to mitigate Gibbs
phenomenon in Fourier–Bessel series solutions of a dynamic sphere problem. Int. J. Numer. Meth. Biomed. Engng.,
26(1276–1292), 2010.
[48] A. B. Kuijlaars, K. T.-R. McLaughlin, W. Van Assche, and M. Vanlessen. The Riemann–Hilbert approach to strong
asymptotics of orthogonal polynomials on [−1, 1]. Adv. Math., 188:337–398, 2004.
[49] J. N. Lyness. Computational techniques based on the Lanczos representation. Math. Comp., 28(125):81–123, 1974.
[50] R. Platte, L. N. Trefethen, and A. Kuijlaars. Impossibility of fast stable approximation of analytic functions from
equispaced samples. SIAM Rev., 53:308–318, 2011.
[51] J. Steinberg. Oblique projections in Hilbert spaces. Integr. equ. oper. theory, 38(1):81–119, 2000.
[52] E. Tadmor. Filters, mollifiers and the computation of the Gibbs’ phenomenon. Acta Numer., 16:305–378, 2007.
[53] E. Tadmor and J. Tanner. Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral
information. Found. Comput. Math., 2(2):155–189, 2002.
26
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement