polynomial frames: a fast tour - uni-luebeck.de · for all polynomials rof degree at most 2n [42,...

Polynomial Frames: A Fast Tour

H. N. Mhaskar and J. Prestin

Abstract. We present a unifying theme in an abstract setting forsome of the recent work on polynomial frames on the circle, the unitinterval, the real line, and the Euclidean sphere. In particular, wedescribe a construction of a tight frame in the abstract setting, so thatcertain Besov approximation spaces can be characterized using theabsolute values of the frame coefficients. We discuss the localizationproperties of the frames in the context of trigonometric, Jacobi, andspherical polynomials, and discuss some applications.

§1. Introduction

Wavelet analysis is perhaps one of the fastest growing areas of analysis.Some of the applications include image processing, signal processing, timeseries analysis, probability density estimation, neural networks, data com-pression, etc. Traditionally [4, 6, 39], wavelets are defined using the notionof multiresolution analysis. Let φ ∈ L2(R), φj,k := 2j/2φ(2j ◦ −k), andVj be the closure in L

2(R) of the linear span of the functions {φj,k}k∈Z.The function φ is said to generate a multiresolution analysis (cf. [39, Def-inition 7.1]) if {φ0,k} forms an orthonormal basis for V0, and each of thefollowing conditions is satisfied:

(a) Vj ⊆ Vj+1, j ∈ Z,

(b) L2-closure

(

⋃

j∈Z Vj

)

= L2(R),⋂

j∈Z Vj = {0},

(c) f(x) ∈ Vj iff f(2x) ∈ Vj+1, j ∈ Z.

The function φ is called a scaling function, and the spaces Vj are calledscaling spaces. For each integer j, the wavelet space Wj is defined to bethe orthogonal complement of Vj in Vj+1. Under these conditions on φ,there exists a mother wavelet, ψ ∈ L2(R) such that for each j ∈ Z, thefunctions {ψj,k = 2j/2ψ(2j ◦ −k)}k∈Z form an orthonormal basis for the

Approximation Theory XI: Gatlinburg 2004 101C. K. Chui, M. Neamtu, and L. L. Schumaker (eds.), pp. 101–132.

Copyright Oc 2004 by Nashboro Press, Brentwood, TN.

ISBN 0-0-9728482-5-8

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102 H. N. Mhaskar and J. Prestin

space Wj (cf. [6, Theorem 5.1.1]). It is easy to see that for every functionf ∈ L2(R), one has the L2-convergent expansion

f =∑

j,k∈Z〈f, ψj,k〉ψj,k,

where 〈◦, ◦〉 denotes the usual inner product.Typically, the coefficient cj,k(f) := 〈f, ψj,k〉 reflects the behavior of f

near the point k/2j . For example, let f(x) :=√

| cosx|. The Daubechieswavelet coefficients of f are large only near the “bad points” π/2 and3π/2; while the Fourier coeffcients reveal no such information localized inthe time domain (cf. [6, Chapter 9]). Nevertheless, it is well known thatthe sequence of Fourier coefficients of a 2π-periodic function, integrableon [0, 2π], uniquely determine the function, and hence, all its features.Therefore, it is an interesting question to develop an expansion of theform

∑

j,k cj,k(f)ψj,k, where ψj,k are preferably trigonometric polynomi-als, the coefficients cj,k are computed using the Fourier coefficients of f ,and reflect local features of f . The problem appears in many differentcontexts; for example, the problem of hidden periodicities [33], directionfinding in phased array antennas [29, 62], and solution of partial differen-tial equations [8, 9, 18, 19]. Similar questions can be studied in the contextof orthogonal polynomial expansions, spherical harmonic expansions, etc.

Apart from time-frequency localization, some other considerations thatare usually used in the choice of a wavelet basis are [39, Section 7.2]: num-ber of vanishing moments, the size of the support of the mother waveletand scaling functions (or their rate of decay near infinity), smoothness ofthe scaling function and the mother wavelet, etc. These requirements areusually conflicting with each other. Thus, a compactly supported motherwavelet cannot have all its moments vanishing; there is no infinitely oftendifferentiable orthonormal wavelet with bounded derivatives that decaysexponentially near infinity [6, Corollary 5.5.3]. For this reason, it has be-come a practice to relax several of the requirements of the definition ofthe multiresolution analysis. For example, the requirement that {φj,k}form an orthonormal system has been dropped in the construction of theChui-Wang spline wavelets [4].

In [5], we proposed a class of trigonometric polynomial wavelets byallowing each Wj to have its own mother wavelet. Our wavelets are them-selves trigonometric polynomials, have some time-frequency localization(corresponding to the Riemann localization principle [80, Chapter II, The-orems 6.3, 6.6]), and most interestingly, the number of vanishing (trigono-metric polynomial) moments increases to infinity as the level j of thewavelets (corresponding to the degree of the trigonometric polynomialsinvolved) increases to infinity.

An effort to find an analogous theory of Hermite polynomial expansionson the whole line does not succeed. There is no nontrivial polynomial P

Polynomial Frames 103

of degree at most 2n+1 and a system of real numbers zk, k = 1, · · · , 2n,such that

∫ ∞

−∞P (t− zk)R(t) exp(−t2)dt = 0

for all polynomials R of degree at most 2n [42, Theorem 11.3.1]. It seemsthat the requirement that the wavelet space be spanned by the translatesof a fixed function has to be abandoned in order for such an effort to suc-ceed. In the context of Jacobi polynomials, there is a modified notion oftranslation (cf. [1]), leading to a convolution structure. The idea of usingsuch “translations” yields a suitable wavelet basis for an arbitrary orthog-onal polynomial expansion [10], even though the convolution structure isnot necessarily available in the general case.

In this paper, we survey certain results which have developed theselines of thought further in the context of trigonometric polynomials, or-thogonal polynomials, and spherical polynomials. There are many sim-ilarities in these theories; in particular, the frames can be developed ina unified manner. We state this unification in Section 2, and apply thisabstract theory in the new context of Freud polynomials, which includethe Hermite polynomials. Although we are not aware of any referenceswhere the theorems in this section appear in the stated form, many ofthe ideas involved can already be found scattered in the known literature.The localization properties of the frames in the context of trigonometric,Jacobi, and spherical polynomials are qualitatively similar, but the de-tails are specific to the individual cases, and are discussed in Sections 3,4, 5 respectively. Some applications are described in Section 6. We donot claim to be exhaustive; we are limited by our subjective interests andcomprehension as well as the page limits.

§2. A General Framework for Frames

In this section, we wish to explore a general setting for frames that seeksto unify the various results available in the theory for trigonometric poly-nomials, orthogonal polynomials (Jacobi polynomials in particular), andspherical polynomials. We will immediately illustrate the results in thecontext of Freud polynomials; the details for the other three cases will bedescribed separately. The Freud polynomials include Hermite polynomialsas a special case. In this paper, Πx will denote the class of all algebraicpolynomials of degree at most x.

Let (X, µ∗) be a σ-finite measure space, ν be a (possibly signed) mea-sure that is either positive and σ-finite, or has a bounded variation on X,|ν| denote ν if ν is a positive measure, and its total variation measure ifit is a signed measure. If A ⊆ X is |ν|-measurable, and f : A → R is


|ν|-measurable, we write

‖f‖ν;p,A :=

{∫

A

|f(t)|pd|ν|(t)}1/p

, if 1 ≤ p


{pk} of polynomials, such that each pk is of degree k, has a positive leadingcoefficient, and (1) is satisfied with φFk (Q;x) = wQ(x)pk(x) in place of φk,and the Lebesgue measure µL on the real line in place of µ

∗. These poly-nomials {pk} are called Freud polynomials. In the case Q(x) = x2/2, thepolynomial pk is the classical orthonormal Hermite polynomial of degreek. The theory of Freud polynomials is quite well developed [42, 73, 34].They are also distinguished from the other three cases because they lacka convolution structure, and their asymptotic behavior is qualitativelydifferent from that of the Jacobi polynomials. In particular, certain lo-calization estimates to be discussed in Sections 3, 4, 5 in the context oftrigonometric, Jacobi, and spherical polynomials respectively, are not yetknown for Freud polynomials.

We resume the discussion in the abstract framework. If f ∈ L1(µ∗; X),we define

f̂(k) :=

∫

X

f(t)φk(t)dµ∗(t), k = 0, 1, · · · . (3)

If I ⊂ R, we define the space of “polynomials” PI to be the L2(µ∗; X)-closed span of {φk}k∈I . In the case when I = [0, y] for some y ≥ 0, wewill abbreviate the notation and write Py rather than P[0,y].

If Nj is an increasing sequence of positive integers, tending to ∞ asj → ∞, we may define the abstract scaling space Vj to be PNj−1 andthe corresponding wavelet space Wj as usual by P[Nj ,Nj+1−1]. As pointedout in the introduction, one does not hope to obtain the mother waveletψj ∈ Wj in general in the sense that the translates of ψj will span thespace Wj . However, motivated by the modified notion of translation forJacobi polynomials (cf. [1]) and its successful use in [10], we are ledto define formally the translation of a function in L1 by t ∈ X by theformula

∑∞k=0 f̂(k)φk(x)φk(t). In the case when X = [−π, π], and {φk}

is an enumeration of the trigonometric monomials {sin kx, cos kx}, thenthis coincides with the translation f(x− t). In the case when X = [−1, 1],and µ∗ = µα,β , α, β ≥ −1/2, Askey and Wainger [1] have shown thatthis modified translation satisfies many of the properties expected of atranslation; for example, the L1 norm is preserved under translation, anda convolution structure can be defined using this translation. Accordingly,we define our scaling function and the mother wavelets in a unified manneras follows.

If h : [0,∞) → R, we define formally

Φ(h, x, t) :=∞∑

k=0

h(k)φk(x)φk(t). (4)

If ν is a (possibly signed) measure on X we define formally,

σ(ν;h, f, x) :=

∫

X

f(t)Φ(h, x, t)dν(t),


-7.5 -5 -2.5 0 2.5 5 7.5

0

1

2

3

-7.5 -5 -2.5 0 2.5 5 7.5

0

1

2

3

4

5

Fig. 1. Two examples for Φ(h, x, 3) where we have taken φk = φFk ((◦)

2/2, ◦),

and on the left h = χ[0,64], and on the right h =23

`

1 + cos π◦64´2χ[0,64].

whenever the integral is well defined. In the remainder of this sectionand the next few sections, we describe several theorems establishing aclose connection among the properties of the function h and those of thekernels Φ and the operators σ (cf. Figure 1).

Let I be a compact subset of [0,∞), hI : R → R be a function suchthat hI(x) = 0, if x 6∈ I, and hI(k) 6= 0 if k is an integer in I. We note thatfor each t ∈ X, Φ(hI , ◦, t) ∈ PI , and for each x ∈ X, Φ(hI , x, ◦) ∈ PI . It isnot difficult to see that σ(µ∗;hI , f, x) =

∑

k∈I hI(k)f̂(k)φk(x), so that hI

acts as a filter with I as the “pass-band”. If we define h[−1]I (x) = hI(x)

−1

if hI(x) 6= 0 and 0 otherwise, it is clear that for P ∈ PI , we have

σ(µ∗;h[−1]I , σ(µ∗;hI , P ), x) = σ(µ

∗;hI , σ(µ∗;h[−1]I , P ), x) = P (x).

Thus, thinking of Φ(hI) as a “mother wavelet” for the space PI , and

σ(µ∗;hI) as the corresponding continuous wavelet transform, Φ(h[−1]I )

plays the role of the dual wavelet, and σ(µ∗;h[−1]I ) is the inverse of thistransform.

The transforms can be discretized easily using quadrature formulas. A(possibly signed) measure ν will be called a Marcinkiewicz–Zygmund (M–Z) quadrature measure of order N if each of the following conditions holds.The measure ν is a positive σ-finite measure, or a signed measure withfinite total variation, every µ∗–measurable function is also |ν|-measurable,

‖T‖ν;p,X ≤M‖T‖µ∗;p,X, T ∈ PN , 1 ≤ p ≤ ∞, (5)

and∫

X

T1(t)T2(t)dν(t) =

∫

X

T1(t)T2(t)dµ∗(t), T1, T2 ∈ PN . (6)


When the support of the measure ν is a finite set, the inequality (5) iscalled an M–Z inequality. In the context of Freud polynomials, an exampleof M–Z quadrature measures is developed in [49, 43] under some additionalconditions.

If f ∈ L2(µ∗; X), the completeness of the system {φk} implies thatf =

∑∞k=0 f̂(k)φk, with the convergence in the sense of L

2(µ∗; X). Now,let {Nj}∞j=0 be an increasing sequence of integers, Nj → ∞ as j → ∞,N0 = 0, and Ij := [Nj , Nj+1 − 1]. For j = 0, 1, · · · , let νj be an M–Zquadrature measure of order Nj+1 − 1. Then it is easy to verify using (6)that for f ∈ L2(µ∗; X),

f =∞∑

j=0

∫

X

σ(µ∗;hIj , f, t)Φ(h[−1]Ij

, ◦, t)dνj(t),

where the convergence is in the sense of L2(µ∗; X). We note that theintegrals are, in fact, sums in the case when each νj is supported on afinite subset of X. The set

{Φ(hIj , ◦, t) : t ∈ supp(νj), j = 0, 1, · · · }

is not, in general, a frame for the whole space L2(µ∗; X). Nevertheless,we have the following theorem to show that each set {Φ(hIj , ◦, t) : t ∈supp(νj)}, j = 0, 1, · · · , is a frame for PIj , even in the sense of Lp norms.Analogues of the following theorem can be found, for example, in [52, 53,47, 48].

Theorem 1. Let 1 ≤ p ≤ ∞, N ≥ 0, I ⊆ [0, N ] be a compact set, µ, νbe M–Z quadrature measures of order N , and M be the larger of the twocorresponding constants appearing in (5). For h : [0,∞) → R for whichΦ(h) defines a µ∗ × µ∗–measurable function on X × X, let

B(h) := M supx∈X

‖Φ(h, x, ◦)‖µ∗;1,X. (7)

Let hI : R → R be a function such that hI(x) = 0, if x 6∈ I, and hI(k) 6= 0if k is an integer in I. Then for f ∈ Lp(µ; X),

‖σ(µ;hI , f)‖ν;p,X ≤ B(hI)‖f‖µ;p,X. (8)

If P ∈ PI thenσ(ν;h

[−1]I , σ(µ;hI , P )) = P, (9)

and we have

(

B(h[−1]I )

)−1‖P‖µ;p,X ≤ ‖σ(µ;hI , P )‖ν;p,X ≤ B(hI)‖P‖µ;p,X. (10)


Proof: The estimate (8) follows easily using the Riesz-Thorin interpola-tion theorem (cf. [44, Lemma 4.1]). The formula (9) is a simple conse-quence of (6), and (10) follows from (9) and (8).

We pause again in the discussion to give an example of bounds on thequantity B(h) in (7) in the context of Freud polynomials.

Theorem 2. Let h = {h(k)} be a sequence with h(k) = 0 if k is greaterthan some positive integer, and wQ be a Freud weight. Then

supx∈R

∥

∥

∥

∥

∥

∞∑

k=0

h(k)φFk (Q;x)φFk (Q; ◦)

∥

∥

∥

∥

∥

µL;1,R

≤ c∞∑

k=0

k|h(k + 2) − 2h(k + 1) + h(k)|, (11)

where c depends only on Q. In particular, if A > 0, h : [0,∞) → R,h(x) = 0 if x ≥ A, and h has a derivative having bounded variation V (h′),then

supn≥1, x∈R

∥

∥

∥

∥

∥

∞∑

k=0

h(k/n)φFk (Q;x)φFk (Q; ◦)

∥

∥

∥

∥

∥

µL;1,R

≤ cAV (h′), (12)

where c depends only on Q.

Proof: It is well known [15] that the Fejér means of the Freud polynomialexpansion are uniformly bounded. Using a summation by parts argumentas in [50, p. 103–104], this fact implies that the operators σ(µL;h, ◦) havenorms that can be estimated by the right hand side of (11). In turn, thesenorms are given by the left hand side of (11).

We resume the discussion in the abstract setting. If one relaxes therequirement that the spaces Wj should form an orthogonal decomposition,or even that Wj ∩Wk = {0} if j 6= k, one can obtain a tight frame for thewhole space L2(µ∗; X), while retaining sufficient flexibility to achieve othersuch goals as Lp-convergence, localization, characterization of smoothnessclasses, construction of bases, etc. An impressive and explicit constructionof tight frames is recently given by Narcowich, Petrushev, and Ward [57]in a very general set up of Hilbert spaces. We present below a slightlydifferent variation of their constructions, that is more in keeping with thecurrent set up and later applications we have in mind.

Theorem 3. Let Nj be an increasing sequence of integers, Nj → ∞ asj → ∞, N0 = 0. For j = 0, 1, · · · , let νj be an M–Z quadrature measure oforder Nj+1−1. Let {gj,k}∞j,k=0 be a matrix of real numbers such that eachof the following conditions is satisfied: (i) For each j = 0, 1, · · · , gj,k = 0


if k ≥ Nj+1, (ii) For each k = 0, 1, · · · ,∑∞

j=0 g2j,k = 1. For j = 0, 1, · · ·

and x, t ∈ X we write

ψj,t(x) :=∞∑

k=0

gj,kφk(x)φk(t). (13)

Let 〈◦, ◦〉 denote the inner product of L2(µ∗; X). Then for f ∈ L2(µ∗; X),

f =∞∑

j=0

∫

X

〈f, ψj,t〉ψj,tdνj(t), (14)

with the convergence in the sense of L2(µ∗; X), and

∞∑

j=0

∫

X

|〈f, ψj,t〉|2dνj(t) = ‖f‖2µ∗;2,X, f ∈ L2(µ∗; X). (15)

Proof: Using (6), we deduce that for each x, u ∈ X and j = 0, 1, · · · ,∫

X

ψj,t(x)ψj,t(u)dνj(t) =∞∑

k=0

g2j,kφk(x)φk(u). (16)

Since gj,k = 0 for k ≥ Nj+1, we may use Fubini’s theorem to calculatethat

∫

X

〈f, ψj,t〉ψj,tdνj(t) =∞∑

k=0

g2j,kf̂(k)φk, (17)

and∫

X

|〈f, ψj,t〉|2dνj(t) =∞∑

k=0

g2j,k|f̂(k)|2.

Since∑∞

j=0 g2j,k = 1 for each k = 0, 1, · · · , these equations imply (14) and

(15).

We remark that in the case when the measures νj are supported onfinite subsets of X, the integrals in the above theorem are only finite sums.In this case, defining

Wj = span {ψj,t : t ∈ supp(νj)} = span {φk : gj,k 6= 0},

we get a multiscale decomposition of L2(µ∗; X), which in addition, isspanned by a tight frame. The overlap amongst the scales, and the num-ber of vanishing moments in each scale clearly depends upon the supports{k : gj,k 6= 0}. If we require that Wj must be the orthogonal complementof Vj in Vj+1, then gj,k = χ[Nj ,Nj+1−1], and the integral expression in (14)is just the corresponding projection of f , as in [10].


Next, we turn our attention to the Lp-theory. In the context of ourconstructions, we will demonstrate the Lp-convergence of the expansion(14), as well as a characterization of Besov (approximation) spaces us-ing the absolute values of the coefficients 〈f, ψj,t〉. We acknowledge thatthe characterization presented in Theorem 4 below is motivated, in part,by a lecture by Petrushev in December, 2003, where similar results wereannounced in the context of approximation on the sphere.

For y ≥ 0 and f ∈ Lp(µ∗; X), we define

Ey,p(f) := infP∈Py

‖f − P‖µ∗;p,X.

The class of all f ∈ Lp(µ∗; X) for which Ey,p(f) → 0 as y → ∞ will bedenoted by Xp := Xp(µ∗; X). To define the Besov (approximation) spaces(cf. [7, Chapter 7.9]), we fix a sequence {Nj} as in Theorem 3, and definea sequence space as follows. Let 0 < ρ ≤ ∞, γ > 0, and a = {an}∞n=0 bea sequence of real numbers. We define

‖a‖ρ,γ :=

{ ∞∑

n=0

2nγρ|an|ρ}1/ρ

, if 0 < ρ 0, and we assume that each νj is an M–Z quadrature measure oforder Nj+2 − 1, that satisfies (5) with the constant M being independentof j. Let 1 ≤ p ≤ ∞ and νj �p µ∗. Further, suppose that

supj≥0

supt∈X

‖Φ(hj , ◦, t)‖µ∗;1,X ≤ c. (19)


Let f ∈ Xp. Then (14) holds in the sense of Xp convergence. We define

τj(f) := σ(νj ;hj , f) − σ(νj−1;hj−1, f), τ∗j (f) := σ(µ∗;hj − hj−1, f).

Let 0 < ρ ≤ ∞, 0 < γ


Hence, part (a) implies part (b). Similarly, part (a) implies part (c). If(b) holds then (23) implies that {‖τj(f)‖µ∗;p,X} ∈ bρ,γ as well. In view of(24),

ENn,p(f) ≤∞∑

j=n

‖τj(f)‖µ∗;p,X.

The discrete Hardy inequality [7, Lemma 3.4, p. 27] now leads to part (a).The proof that (c) implies (a) is similar.

Next, for any M–Z quadrature measure ν of order Nj+1−1, we considerthe operator

Tj(ν; g) :=

∫

X

g(t)ψj,tdν(t).

Since ψj,t(x) = ψj,x(t), using Riesz-Thorin interpolation theorem (cf. [44,Lemma 4.1]) and the condition (20), we deduce that

‖Tj(ν; g)‖ν;p,X ≤ c‖g‖ν;p,X. (25)

We use this estimate with g(t) = 〈f, ψj,t〉 and ν = νj . In view of (17), thisimplies that

‖τ∗j (f)‖νj ;p,X = ‖Tj(νj ; g)‖νj ;p,X ≤ c‖〈f, ψj,◦〉‖νj ;p,X.

Thus, part (d) implies part (c).Since hj(x)−hj−1(x) = 0 if x ∈ [0, Nj−1], we have

∫

XP (t)ψj,tdνj(t) =

0 for P ∈ ΠNj−1 . We choose P such that ‖f − P‖µ∗;p,X ≤ 2ENj−1,p(f).Then (25) with ν = µ∗, together with the fact that νj �p µ∗ imply that

‖〈f, ψj,◦〉‖νj ;p,X = ‖Tj(µ∗; f − P )‖νj ;p,X ≤ cENj−1,p(f).

Therefore, part (a) implies part (d).

We comment about the estimates (19) and (20) in the context of Freudpolynomials. Let wQ be a Freud weight, h be a nonincreasing function,which is equal to 1 if 0 ≤ x ≤ 1/2 and 0 if x ≥ 1. Let Nj = 2j , andhj(x) = h(x/2

j). The estimate (12) in Theorem 2 shows that the esti-mate (19) (respectively, (20)) holds if h is twice (respectively, four times)continuously differentiable on [0, 1].

We note that the characterizations (c) and (d) in the above theorem are

based on the Fourier coefficients {f̂(k)}. If the measures νj are supportedon finite subsets of X, then the characterization (b) uses the values of fat the points in these subsets. An application of such a characterizationto the problem of bit representation of a function on a Euclidean sphereis given in [45].

Finally, we note that when each νj is supported on a finite set, thecondition (19) implies that the set {Φ(hj − hj−1, ◦, t) : t ∈ supp(νj)}


is also a frame for L2(µ∗; X). This can be shown following the proof ofTheorem 3.2 in [45].

For the convenience of the reader, we summarize some parts of ourdiscussion in the case when the measures νj are discrete measures.

Corollary 1. With the notations as in Theorem 3, let each νj be sup-ported on a finite set of points {tj,k}k=1,··· ,Mj in X, with νj({tj,k}) =wj,k > 0. For f ∈ L2(µ∗; X), we have

f =

∞∑

j=0

Mj∑

k=1

wj,k〈f, ψj,tj,k〉ψj,tj,k , (26)

with the convergence in the sense of L2(µ∗; X), and

∞∑

j=0

Mj∑

k=1

wj,k|〈f, ψj,tj,k〉|2 = ‖f‖2µ∗;2,X.

Let f ∈ X∞, gj,k be as in (18), and (20) be satisfied. Then (26) holds inthe sense of X∞–convergence, and f ∈ Bµ∗;∞,ρ,γ if and only if

{

max1≤k≤Mj

|〈f, ψj,tj,k〉|}

∈ bρ,γ .

§3. Trigonometric Polynomials

In this section, we take X = [−π, π], µ∗ to be the normalized Lebseguemeasure, the trigonometric monomials, with the enumeration

{1,√

2 cosx,√

2 sinx,√

2 cos 2x,√

2 sin 2x, · · · },

as the fixed orthonormal set, and omit the mention of X and µ∗ from thenotations. We take Nj so that the scaling spaces are defined by

Vj := span{1, cosx, sinx, · · · , cos 2jx, sin 2jx}.

A typical example of an M–Z quadrature measure of order 2N + 1 isthe measure that associates the mass 1/(3N) with each of the points2πk/(3N), k = 0, · · · , 3N − 1 (cf. [80, Chapter X, Formula (2.5), Theo-rems 7.5, 7.28]). M–Z quadrature measures supported on arbitrary sys-tems of points can be obtained as a special case (q = 1) of Theorem 10below.

The notion of translation as introduced in the previous section is thesame as the translation in the usual sense. Shift-invariant spaces of peri-odic functions have been studied from the perspective of multiresolution


analysis by Koh, Lee, and Tan [28], Plonka and Tasche [64], Narcowichand Ward [58], Goh, Lee, and Teo [23], Chen [3], and Maximenko andSkopina [40], among others. Approximation properties of periodic waveletseries are described by Skopina; e.g., in [74]. Localization estimates forthe obvious and important example of de la Vallée Poussin kernels as wellas algorithmic aspects are discussed in [67, 71].

Clearly, the question of best possible localization of the kernels Φ andψ (cf. (4) and (13)) depends on how we want to measure such localization.

It is easy to prove that the Dirichlet kernel (1/√

2j+1 + 1)

2j∑

k=−2jeik◦e−ikt

is a solution of the problem of maximizing |T (t)| among all T ∈ Vj with‖T‖2 = 1. In this sense, the Dirichlet kernel may be thought of as thebest localized element of Vj . In this section, we review the localizationproperties from two other points of view. We discuss the rate of decay ofthe kernel functions Φ and ψ introduced in Section 2, and also the effectof the choice of the mask on certain uncertainty principles.

3.1. Time domain behavior

We now discuss the conditions to ensure bounds required in (7), (19),and (20), as well as the localization of the kernels Φ and ψ. The forwarddifferences of a sequence {h(k)} are defined recursively as follows:

∆1h(k) := ∆h(k) := h(k + 1) − h(k), ∆rh(k) = ∆r−1(∆h(k)), r ≥ 2,

and we find it convenient to write ∆0h(k) := h(k).The following theorem is proved in [52, pp. 104–105].

Theorem 5. Let {h(k)} be a bi-infinite sequence.(a) If 1 ≤ p ≤ ∞ and ∑k∈Z |k|2−1/p|∆2h(k)| < ∞ then

∑

k∈Z h(k)eik◦

converges in the Lp norm, and we have∥

∥

∥

∥

∥

∑

k∈Zh(k)eik◦

∥

∥

∥

∥

∥

p

≤ c∑

k∈Z|k|2−1/p|∆2h(k)|. (27)

(b) Let K ≥ 0 be an integer, and ∑k∈Z |h(k)|


If h is a K times iterated integral of a function h(K) having a boundedvariation V (h(K)), then (28) implies that for x ∈ [−π, π] \ {0}, and n =1, 2, · · · ,

∣

∣

∣

∣

∑

k∈Zh(|k|/n)eikx

∣

∣

∣

∣

≤ cV (h(K))

nK |x|K+1 .

The estimates for the boundedness and localization of the kernels ψ canalso be obtained from (27) and (28) for sufficiently smooth functions h inthe same way as in Corollary 2. The estimates in Theorem 5 have beenused in [54] to obtain a characterization of local Besov spaces, similarto Theorem 4. The characterization in terms of the tight frame are notgiven there, but can be derived by a trivial modification of the proofsthere. A trigonometric polynomial kernel having exponential decay hasbeen developed in [51] using some complex analysis ideas from the work[17] of Gaier.

3.2. Uncertainty principles

To state an uncertainty principle for functions in L2, the necessary notionsof variance for periodic functions need to be introduced. We refer toBreitenberger [2] and [58, 69, 70, 22]. For a function f ∈ L2, representedby its Fourier series; i.e., f(x) =

∑∞s=−∞ cse

isx, the first trigonometricmoment is defined by

τ(f) :=1

2π

∫ 2π

0

eix|f(x)|2 dx =∞∑

s=−∞cscs+1, (29)

and the angle variance is defined by

varA(f) :=‖f‖42 − |τ(f)|

2

|τ(f)|2=

∣

∣

∣

∣

∣

∑∞s=−∞ |cs|2

∑∞s=−∞ cscs+1

∣

∣

∣

∣

∣

2

− 1,

The frequency variance for f ∈ L2 is defined by

varF (f) := ‖f ′‖22 +〈f ′, f〉2

‖f‖22=

∞∑

s=−∞s2|cs|2 −

(

∑∞s=−∞ s|cs|

2)2

∑∞s=−∞ |cs|

2 .

Note that the variances attain the value ∞ if and only if τ(f) = 0 orf ′ 6∈ L2, respectively.

In the following theorem [2, 58, 69], an uncertainty relation for L2 isformulated, which only excludes single frequency functions of the formceik◦, c ∈ C, k ∈ Z. In this case, varF = 0 and varA = ∞.


Theorem 6. For functions f ∈ L2 which are not of the form ceik◦, c ∈ C,k ∈ Z, we have

U2π(f) :=

√

varA(f)varF (f)

‖f‖2>

1

2,

where the constant 1/2 is the best possible.

We recall that if H is a sufficiently rapidly decaying and smooth func-tion on R, then the Heisenberg uncertainty product UR(H) ≥ 1/2, withequality only for the Gaussians [4, Theorem 3.5]. If H is a sufficientlyrapidly decaying and smooth function on R, and ha(x) := H(x/a), thenthe kernels Φ(ha, x, t) =

∑∞k=−∞ ha(k)e

ikxe−ikt satisfy [70]

lima→∞

U2π(Φ(ha, ◦, t)) = UR(H) −1 by

wα,β(x) :=

{

(1 − x)α(1 + x)β , if x ∈ (−1, 1),0, if x ∈ R \ (−1, 1).

The corresponding measure µα,β is defined by dµα,β(x) := wα,β(x)dx, andwe will simplify our notations by writing α, β in place of µα,β; for example,we write ‖f‖α,β;p,A instead of ‖f‖µα,β ;p,A. In this section, we take φn to bethe orthonormalized Jacobi polynomial pn

(α,β) [14, Chapter 1, Section I.8].Nevai [60, Theorem 25, p. 168] has given an example of M–Z quadra-

ture measures for the Jacobi weights. For m ≥ 1, let {xk,m}mk=1 be the


zeros of p(α,β)m , and

λk,m :=

{

m−1∑

r=0

p(α,β)r (xk,m)2

}−1

, k = 1, · · · ,m.

Nevai has proved that for m ≥ cn, the measure ν∗m that associates themass λk,m with each xk,m is an M–Z quadrature measure of order n.

Several constructions as well as algorithmic questions in the contextof Jacobi polynomials are discussed in [26, 78, 65, 11]. In the context ofthe theory in Section 2, polynomial frames using Jacobi polynomials weredeveloped for the problem of detection of singularities in [53]. The workstarted there was continued in [44], where local analogues of Theorem 4were obtained. Our main objective in this section is to discuss the esti-mates and localization properties of the kernels Φ and ψ (cf. (4) and (13))in Section 2 from the two points of view as in the previous section.

4.1. Time domain behavior

The analogue of Theorem 5 is the following theorem in [44].

Theorem 7. Let α, β ≥ −1/2.(a) Let h = {h(k)} be a sequence with h(k) = 0 if k is greater than somepositive integer, and K > max(α, β) + 3/2 be an integer. Then

supx∈[−1,1]

∥

∥

∥

∥

∥

∞∑

k=0

h(k)pk(α,β)(x)pk

(α,β)

∥

∥

∥

∥

∥

α,β;1,[−1,1]

≤ cK

∑

j=1

∞∑

k=0

(k+1)j−1|∆jh(k)|.

In particular, if h : [0,∞) → R is a compactly supported function, h is aK times iterated integral of a function h(K) having bounded variation on[0,∞), V (h(K)) denotes the total variation of h(K), then

supn≥1, x∈[−1,1]

∥

∥

∥

∥

∥

∞∑

k=0

h(k/n)pk(α,β)(x)pk

(α,β)

∥

∥

∥

∥

∥

α,β;1,[−1,1]

≤ cV (h(K)). (30)

(b) Let δ > 0, D ≥ 0, K ≥ D+α+β+2 be an integer, and h : [0,∞) → Rbe a function which is a K times iterated integral of a function of boundedvariation, h′(x) = 0 if 0 ≤ x ≤ δ, and h(x) = 0 if x > c. Then forx, x0 ∈ [−1, 1], η > 0, and |x− x0| ≤ η/2,

supn≥1, t∈[−1,1]\[x0−η,x0+η]

nD

∣

∣

∣

∣

∣

∞∑

k=0

h(k/n)pk(α,β)(x)pk

(α,β)(t)

∣

∣

∣

∣

∣

≤ c.

First, we comment about the estimates (19) and (20) in the context ofJacobi polynomials. Let α, β ≥ −1/2, K > max(α, β)+3/2 be an integer,


and h be a nonincreasing function, which is equal to 1 if 0 ≤ x ≤ 1/2and 0 if x ≥ 1. Let Nj = 2j , and hj(x) = h(x/2j). The estimate (30) inTheorem 7(a) shows that the estimate (19) (respectively, (20)) holds if his a K+1 (respectively, 2K+2) times continuously differentiable on [0, 1].

Concerning the localization of the kernel ψ, Theorem 7(b) implies,in particular, the following. Let K > α + β + 2 be an integer, and hbe a nonincreasing function, which is equal to 1 if 0 ≤ x ≤ 1/2 and 0if x ≥ 1. Let Nj = 2j , and hj(x) = h(x/2j). If h is 2K + 2 timescontinuously differentiable function on [0, 1] then for 0 ≤ D ≤ K−α−β−2,x, x0 ∈ [−1, 1] and η > 0,

supn≥1, t∈[−1,1]\[x0−η,x0+η]

nD|ψj,t(x)| < c, |x− x0| ≤ η/2.

We find it worthwhile to quote the following Lemma 4.10 from [44] inthis context.

Theorem 8. Let α, β ≥ −1/2, K ≥ 1 be an integer, h(k) = 0 for allsufficiently large k. Then

∣

∣

∣

∣

∣

∞∑

k=0

h(k)pk(α,β)(1)pk

(α,β)(t)

∣

∣

∣

∣

∣

≤ c

∞∑

k=0

min

(

(k + 1)2,1

1 − t

)α/2+K/2+1/4

×K−1∑

m=0

(k + 1)α+1/2−m|∆K−mh(k)|, if 0 ≤ t < 1,∞∑

k=0

(k + 1)α+β+1K−1∑

m=0

(k + 1)−m|∆K−mh(k)|, if − 1 ≤ t < 0,

and a similar estimate holds with −1 replacing 1 in the leftmost expressionabove.

Figure 2 illustrates the localization of the frame elements ψj,t, where

gj,k =

{

(8/3) sin4(kπ/2j), if k = 2j , · · · , 2j+1,0, otherwise.

(31)

4.2. Uncertainty principles

An uncertainty principle for Legendre polynomials can be deduced fromthe more general results by Narcowich and Ward in [59]. The uncertaintyprinciples and their sharpness for ultraspherical expansions are proved inthe fundamental paper of Rösler and Voit [77]. Uncertainty principles in


-1 -0.5 0 0.5 1-20

-10

0

10

20

-1 -0.5 0 0.5 1

-100

0

100

200

300

400

Fig. 2. With α = β = 0, gj,k as in (31), we plot ψ6,0 on the left, and ψ4,1 onthe right, which already shows a good localization.

the case of arbitrary Jacobi expansions are recently given by Li and Liu[36]. Analogous to (29), they have defined

τα,β(f) :=1

2α+ 2

∫ 1

−1((α+ β + 2)x+ α− β)|f(x)|2dµα,β(x),

and the frequency variance by

varF,α,β(f) :=∞∑

n=1

n(n+ α+ β + 1)|f̂(n)|2,

where f̂(n) denotes the Fourier coefficient in the orthonormalized Jacobiseries. Their theorem can be reformulated as follows.

Theorem 9. Let α, β > −1 and ‖f‖α,β;2,[−1,1] = 1. Then

(1−τα,β(f))(

β + 1

α+ 1+ τα,β(f)

)

varF,α,β(f) ≥1

4(α+β+2)2τα,β(f)

2. (32)

Further, let h�(n) := A� exp(−n(n + α + β + 1)�), where A� is chosen sothat ‖Φ(h�, 1, ◦)‖α,β;2,[−1,1] = 1. Then

lim�→0+

� varF,α,β(Φ(h�, 1, ◦)) = (α+ 1)/2,

lim�→0+

�−1(1 − τα,β(Φ(h�, 1, ◦))) = (α+ β + 2)/2,

and hence, the inequality (32) is sharp.


§5. Spherical Polynomials

In this section, let q ≥ 1 be a fixed integer, X = Sq denote the unitsphere of the Euclidean space Rq+1, µ∗ = µ∗q be the volume element ofSq, and ωq = µ

∗q(S

q). For a fixed integer ` ≥ 0, the restriction to Sqof a homogeneous harmonic polynomial of degree ` is called a sphericalharmonic of degree `. The class of all spherical harmonics of degree ` willbe denoted by Hq` , and its dimension is given by

d q` := dim Hq` =

2`+ q − 1`+ q − 1

(

`+ q − 1`

)

, if ` ≥ 1,1, if ` = 0.

If we choose an orthonormal basis {Y`,m : m = 1, · · · , dq`} for each Hq` ,

then the set {φk} = {Y`,m : ` = 0, 1, . . . and m = 1, · · · , d q`} is an or-thonormal basis for L2(Sq). Here, we choose an enumeration such thatdeg φk < deg φ` implies k < `. With

P`(q + 1; ◦) :=√

ωqωq−1d

q`

p`(q/2−1,q/2−1),

one has the following well known addition formula [55]. For x,y ∈ Sq,d q∑̀

k=1

Y`,k(x)Y`,k(y) =d q`ωq

P`(q + 1;x · y), ` = 0, 1, · · · . (33)

For any x ≥ 0, the class of all spherical harmonics of degree ` ≤ x will bedenoted by Πqx. For any integer n ≥ 0, Πqn =

⊕n`=0 H

q` , and it comprises

the restriction to Sq of all algebraic polynomials in q + 1 variables oftotal degree not exceeding n. Further information regarding sphericalharmonics can be found, for example, in [55, 63, 12].

The connection between M–Z inequalities and quadrature formulas wasnoted in [47], where we proved the existence of M–Z quadrature measuresfor “scattered data” on the sphere. More specifically, we proved the fol-lowing.

Theorem 10. There exist constants αq and α̃q with the following prop-erty. Let C be a finite set of distinct points on Sq,

δC := supx∈Sq

dist(x, C),

and n be an integer with α̃q ≤ n ≤ αqδ−1C . Then there exists a nonnegativeM–Z quadrature measure νC of order equal to dim(Πqn), supported on asubset C1 of C with

|C1| ∼ nq ∼ dim(Πqn).Further, δC ≤ δC1 ≤ cδC and minξ,ζ∈C1,ξ 6=ζ dist(ξ, ζ) ≥ c1δC .


A sharper version of Theorem 10 with lower bounds for the nonzeroweights is recently given by Narcowich, Petrushev, and Ward in [57].

There are many constructions for wavelets and frames on the sphere.For example, we mention the extensive work of the group of Freeden (e.g.,[12, 13]) and that of Skopina and her collaborators (e.g., [76, 24]). Inparticular, ideas similar to those in Section 2, but involving two kernelsψ and ψ̃ can already be found in their papers. A few other references aregiven in [46, 48].

Our own work was motivated by that of Potts, Steidl, and Tasche [66],where tight frames are constructed in the context of S2 using midsections ofthe Fourier expansions and a grid of points on the sphere, and a numberof algorithmic questions are discussed. An analogue of Theorem 1 wasproved in [46], where some applications to the problem of the detection ofsingularities on the sphere were also discussed.

In view of (33), it is convenient to define the kernels Φ of (4) in termsof the corresponding univariate kernel for Jacobi polynomials. Let

ΦS(h, x) := ω−1q−1

∞∑

`=0

h(`)p`(q/2−1,q/2−1)(1)p`

(q/2−1,q/2−1)(x)

=∞∑

`=0

h(`)dq`ωq

P`(q + 1;x).

Defining a function h̃ from h to correspond to the enumeration of {Y`,m},the kernel Φ(h̃,x,y) of (4) in this context is given by ΦS(h,x · y), x,y ∈S

q. Consequently, the bounds on the norms on these kernels (as well asthe corresponding kernels ψj,y) can be deduced easily from Theorem 7,and the bounds for the localization of these kernels can be deduced fromTheorem 8. Another proof of these bounds in a slightly different ansatzis given in [57], where the authors develop tight frames in the setting ofa general Hilbert space, and discuss the localization and Lp convergenceproperties for the sphere.

In the lectures of Petrushev in December, 2003 and Narcowich in May,2004 [56, 57], results similar to Theorem 4 were announced. A similarcharacterization of local Besov spaces is given in [45]. Although the resultsin [45] are not formulated in terms of the coefficients of a tight frame, itis not difficult to modify the proof to include this case.

Uncertainty principles for spheres were first introduced for S2 by Nar-cowich and Ward in their fundamental paper [59]. The uncertainty prin-ciples in the more general case of Sq and their sharpness were studied byRösler and Voit [77]. The relation of the optimality of the uncertaintyproduct and the smoothness of the mask function is discussed by Gohand Goodman in [22]. Extremal functions for the time variance have beenshown in [46] and [32] to be multiples of the corresponding ultrasphericalpolynomials.


§6. Applications

Applications of polynomial wavelets and frames include numerical solu-tions of pseudo-differential operator equations on the sphere [35], directionfinding in phased array antennas [54], representation of functions on thesphere using a near-minimal number of binary digits [45], and constructionof zonal function network frames on the sphere [48]. An application to thenumerical simulation of turbulent channel flow is presented by Fröhlichand Uhlmann in [16]. In this paper, we describe in detail an applicationof the ideas to the problem of finding Schauder bases for the spaces ofcontinuous functions, and to the problem of detection of singularities.

6.1. Bases

We note that elements of a frame are not necessarily linearly independent,and hence, a frame expansion may not be unique. In contrast, a set {fj} ofa Banach space Y is called a Schauder basis for Y if the linear combinationsof elements of this set are dense in Y , and each element f of Y has a uniquerepresentation of the form

∑

j cj(f)fj .We consider in some detail the case when X is a connected real interval,

µ∗ is a mass distribution; i.e.,∫

X|t|ndµ∗(t) < ∞ for n = 0, 1, · · · , and let

φk be the polynomial of degree k, such that {φk} is an orthonormal systemwith respect to µ∗.

We begin by exploring a choice of the functions ψj,t defined in (13)with

gj,k

{

6= 0, if 2j < k ≤ 2j+1,= 0, otherwise.

(34)

Using the ideas in [10], it is not difficult to prove the following result.

Theorem 11. Let tj,s, s = 1, . . . , 2j , be the zeros of φ2j . Then the

polynomials ψj,tj,s , s = 1, . . . , 2j , are linearly independent. In particular

span{

ψj,tj,s : s = 1, . . . , 2j}

= span{

φk : k = 2j + 1, . . . , 2j+1

}

.

In view of (34), we conclude immediately from this theorem the or-thogonal decomposition

L2(µ∗; X) = L2-closure

span{φ0} ⊕∞

⊕

j=0

span{

ψj,tj,s : s = 1, . . . , 2j}

.

However, this is not a full orthogonal decomposition of L2(µ∗; X), becauseψj,tj,r and ψj,tj,s are in general not orthogonal to each other. We are aware

of only one such example, with gj,k ∈ {0, 1}. Let φk := p(12, 12 )

k , and tj,s be


the zeros of φ2j . It can be shown that (cf. [10, Lemma 5.25])

1∫

−1

ψj1,tj1,r (x)ψj2,tj2,s(x)√

1 − x2 dx = cj1,rδj1,j2δr,s.

Clearly, {φk} is an orthonormal basis for L2(µ∗; X), but it is not alocalized basis. In search for a localized orthogonal basis for L2(µ∗; X) ina more general setting, we have to change the definition of the generalizedtranslates a little bit. For this purpose, let N ≥ 3M be positive integers,and for s = 0, . . . , 2M − 1, let

ψs(x) :=2M∑

k=−2Mg

(

k

M

)

cos

(

(k −M)(2s+ 1)4M

π

)

φN−M+k(x).

The special cosine symmetry structure yields the following orthogonalityresult ([21]).

Lemma 1. Let g : R → R, supp g ⊆ [−2, 2], g(0) = 1, and

g2(1 + x) + g2(1 − x) = g2(−1 + x) + g2(−1 − x) = 1 (35)

for all x ∈ [0, 1]. Then∫

X

ψr(x)ψs(x)dµ∗(x) = 0, if r 6= s.

We observe that a simple example of the function g satisfying the condi-tions of Lemma 1 is given by

g(x) =

1, if |x| < 1,√2

2 , if |x| = 1,0, otherwise.

Indeed, taking a sequence of the polynomials ψs defined with this g, forexample, with N = 4M = 2j , j = 2, 3, . . . , one obtains together with φ0and φ1 an orthogonal basis of L

2(µ∗; X) with localized elements ψs (cf.Figure 3).

A modification of the ideas developed so far in connection with framessometimes gives us an orthonormal basis for the space C[−1, 1] of contin-uous functions on [−1, 1], equipped with the supremum norm, as usual.A brief introduction to this subject can be found in [37, Chapter 5.4]. Inparticular, the motivation for the results in the remainder of this section isgiven by the following theorem of Privalov [37, Theorem 4.5, Chapter 5].

Theorem 12. Let {nk} be an increasing sequence of integers, and let{Pk ∈ Πnk} be a Schauder basis for C[−1, 1]. Then there exist �, k0 > 0such that nk ≥ (1 + �)k, k ≥ k0.


A similar theorem is proved by Privalov also for the case of periodicfunctions. In 1992, Offin and Oskolkov [61] used Meyer wavelets to con-struct an orthogonal basis for the space of continuous 2π-periodic func-tions, consisting of trigonometric polynomials of a near optimal order. Acomplete solution was given by Lorentz and Sahakian [38] in the case oftrigonometric polynomials. In [72], we presented another construction,based more on the lines described in this paper, where near optimal ratesfor approximation by partial sums of the basis expansion were proved.

In the case of functions defined on the unit interval, Skopina [75], andindependently, Woźniakowski [79] obtained the following result.

Theorem 13. For any � > 0, there exists a sequence {Pk ∈ Π(1+�)k}where the Pk’s are orthonormal on [−1, 1] with respect to the Lebesguemeasure, and constitute a Schauder basis for C[−1, 1].

To the best of our knowledge, the problem remains open in the casewhen orthonormality is required with respect to Jacobi weights. However,a solution was given in the case of α = β = −1/2 in [27, 21] and in the caseα = ±1/2, β = ±1/2 by Girgensohn in [20]. We describe these results insome detail, since the constructions are similar in spirit to those describedfor frames.

Theorem 14. Let α, β = ±1/2, ε > 0, and η be a positive integer forwhich 3/2η ≤ ε. Let g be a twice iterated integral of a function of boundedvariation, supp g ⊆ [−2, 2], and g(0) = 1. We assume further that gsatisfies the symmetry properties (35), and define

g̃(x) =

0, if x ≤ −3/2,g(2x+ 1), if − 3/2 ≤ x ≤ −1/2,1, if − 1/2 ≤ x ≤ 0,g(x), if x ≥ 0.

Let the sequence of polynomials {Pm}m∈N0 be defined as follows. Form ≤ 2η, set P0 := p0(α,β) and Pm := pm(α,β) for m > 0. For m > 2η,let q ∈ N, r ∈ {1, . . . , 2η−1}, and k ∈ {0, . . . , 2q+1 − 1} be the uniquenumbers such that m = (2η + 2r − 2) 2q + 1 + k. We set M := 2q,N := (2η + 2r) 2q, θk :=

2k+14M π, and define

Pm := M− 1

2

2M∑

s=−2Mg̃

(

sM

)

cos((s−M) θk) pN−M+s(α,β), if r = 1,

Pm := M− 1

2

2M∑

s=−2Mg

(

sM

)

cos((s−M) θk) pN−M+s(α,β), if r > 1,

Then each Pm ∈ Π(1+�)m, and the sequence {Pm} is a Schauder basis for(C[−1, 1], ‖ ◦ ‖α,β;∞), and orthonormal with respect to the measure µα,β.


-1 -0.5 0 0.5 1

-4

-2

0

2

4

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Fig. 3. The polynomial Pm of Theorem 14 for β = α = −1/2, m = 226, andε = 3/4; i.e., Pm ∈ Π287 (left) and ε = 3/64; i.e., Pm ∈ Π229 (right).

6.2. Detection of singularities

We say that a 2π-periodic function f : R → C is piecewise R-smooth, ifthere exist points −π = y0 < y1 < · · · < ym = π such that the restrictionof f to (yi, yi+1), 0 ≤ i ≤ m − 1, is R times continuously differentiable,and f (r)(yi+) and f

(r)(yi+1−) exist as finite numbers for 0 ≤ r ≤ R.Here, ym and y0 are identified as usual. A typical example of a piecewiser + 1-smooth function is given by

Γr(x) =∑

k∈Z, k 6=0

eikx

(ik)r+1, x ∈ R,

which is a 2π-periodic function with r continuous derivatives at each pointon [−π, π] \ {0}, and a singularity of order r at 0. Indeed, any piecewiseR-smooth function can be expressed in a cannonical form as follows:

f(x) =R

∑

r=0

mr∑

j=1

ωj,rΓr(x− yj) + F (x), x ∈ R,

where ωj,r ∈ C, and F is an R times continuously differentiable 2π-periodicfunction on R. The problem of detection of singularities consists of findingthe quantities yj and ωj,r, given the Fourier coeffcients of f . Because ofa number of applications of this problem, some of which are mentioned inthe introduction, many authors have developed algorithms for detectionof singularities; e.g., [33, 8, 9, 30, 18, 19, 31]. In [52], we studied system-atically the performance of different filters with respect to their ability ofdetecting singularities. In the context of trigonometric polynomial frames,we proved the following, where the notations are as in Section 3.

Theorem 15. Let K ≥ 0 be an integer, g be a K times iterated integralof a function of bounded variation, g(x) > 0 if x ∈ (1, 2), and g(x) = 0 if


x 6∈ [1, 2], and let

ψn(x) :=∑

n≤|k|


0 0.5 1 1.5 20

0.002

0.004

0.006

0.008

0 0.5 1 1.5 2

-0.1

-0.05

0

0.05

0.1

Fig. 4. The trigonometric frame transform ofp

| cosx| with polynomials of

degree 32 on the left, the Daubechies wavelet transform (ψD3 )4 ∗ f on the right.The x-axis represents [0, 2π] in multiples of π.

§7. Conclusions

We have discussed the question of obtaining local information about func-tions, starting from global data in the form of Fourier coefficients. Wehave described frames based on the coefficients of a general orthogonalsystem defined on a general measure space, and applied the abstract the-ory to obtain new constructions on the real line using Freud polynomials.Applications to frames consisting of trigonometric, algebraic, and spher-ical polynomials are discussed. In turn, we describe the applications ofthese in the construction of Schauder bases, and the detection of singular-ities. The frame elements discussed here are themselves polynomials. Theframes are also “built up”. Thus, instead of projecting the function firston a very large scaling space, we start with a small amount of data, andconstruct higher and higher order frames with a larger and larger amountof data. In most cases, our constructions can be discretized so as to utilizesamples of the function at scattered sites; i.e., when there is no controlover where to collect the data.

Acknowledgments. The research of first named author was supported,in part, by grant DMS-0204704 from the National Science Foundation andgrant W911NF-04-1-0339 from the U.S. Army Research Office. We wouldlike to thank Carl de Boor, Charles Chui, Tim Goodman, Seng LuanLee, Fran Narcowich, Eitan Tadmor, Manfred Tasche, Maria Skopina,and Joe Ward for their input. We are grateful to Fran Narcowich, PenchoPetrushev, and Joe Ward for giving us their manuscript [57].

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H. N. MhaskarCalifornia State University, Los AngelesLos Angeles, California, 90032, U. S. [email protected]

www.calstatela.edu/faculty/hmhaska

J. PrestinUniversity of LübeckWallstraße 40, 23560, Lübeck, [email protected]

www.math.uni-luebeck.de/prestin

polynomial frames: a fast tour - uni-luebeck.de · for all polynomials rof degree at most 2n [42,...

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