LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS. 1. Introduction

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1 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS Abstract. The local behavior of regular wavelet sampling expansions is quantified. The term regular refers to the decay properties of scaling functions ϕ of a given multiresolution analysis. The regularity of the sampling function corresponding to ϕ is proved. This regularity is used to determine small intervals of sampling points so that the sampled values of a signal f at this finite set of points gives rise to a sampling expansion approximating f to within a predetermined margin of error.. Introduction.. Background. In 99, two of the authors began discussing local sampling for Gabor expansions in the spirit of local sampling results by Helms and Thomas [HT6] and Jagerman [Jag66] from the 96s, cf., Logan [Log84]. These discussions and results were formulated in terms of the Classical Sampling Theorem, see Theorem.. Since then there has been a plethora of activity in the sampling and wavelet communities. For example, there are biennial Sampling Theory and Applications conferences, as well as edited volumes on sampling, e.g., [BF], [Mar], [BZ3]. Also in recent years classical ideas from interpolation theory, e.g., [Sch46], [Wal94], have coalesced with the concept of multiresolution analysis from wavelet theory in a natural structurally appealling way, e.g., see Theorem.6 which is formulated in terms of the mixed norm space C, (Z d, T d ) defined in Definition.. As such, our original idea from 99 has focused on integrating these two directions. The main result of this paper, Theorem 4., is a local sampling theorem in the setting of a subspace of C, (Z, T), in which natural decay conditions are exploited. Before describing the local sampling problem, we shall state the Classical Sampling Theorem (Theorem.). In order to do this we formally define the Fourier transform f of a complex valued function f defined on the real line R to be γ R, f(γ) = f(t)e πitγ dt, where integration is over R and R designates R considered as a frequency domain. We shall be dealing with the usual L p (R) spaces as defined, for example, in [Ben97], [Kat68]; and Mathematics Subect Classification. 4A6, 4A38, 4C4, 4C99, 65G99. Key words and phrases. local sampling and error, MRA and Gabor systems, mixed norm spaces. The second named author gratefully acknowledges support from NSF-DMS Grant (-5) and the General Research Board of the University of Maryland.

2 N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS P W Ω in Theorem. is the Paley-Wiener space P W Ω = {f L (R) : supp f [ Ω, Ω]}, where supp f denotes the support of f. τ y is the translation operator defined as (τ y f)(x) = f(x y). Theorem.. Let T, Ω > satisfy the condition that < T Ω, and let s P W /(T ) satisfy the condition that ŝ S = on [ Ω, Ω] and S L ( R). Then (.) f P W Ω, f = T f(nt )τ nt s, n= where the convergence in (.) is in L (R) norm and uniformly in R. See [Ben97] (Theorem 3..) and [BF] (Chapter ) for a proof and history, respectively... Problem and outline. We shall examine the local behavior of the sampling series (.). In other words, and in the more general case of a scaled version of (.), we shall try to solve the following problem : given m N, an interval I, and an ε >, find the least possible length of the interval [N b m, N c m ] containing I, such that (.) sup x I where N b, N c Z. N c f(x) f(n/ m )s( m x n) n=n b < ε, Our setting is restricted to sampling functions s derived in a natural way from a given multiresolution analysis and scaling function ϕ, see Section. In Theorem 4. of Section 4 we give an estimate of the error formulated in (.) for sufficiently regular sampling functions s. Refinements of this error for a special case is the content of Corollary 4.3. Theorem 4.4 provides the error estimate (.) using the method of Theorem 4., but only assuming decay of a scaling function ϕ of a given multiresolution analysis. The relation between the decay of ϕ and s is established in Theorem 3.6 of Section 3. Finally, in Section 5, we present a local sampling theorem based on the Gabor transform. The results in Sections 3-5 are proved in the setting of the real-line R. It is meaningful to prove analogous results on R d, but the present methods would have to be extended and combined with an intricate geometric analysis. In a sense the error estimated in Section 4 is a truncation error. This error has been extensively studied, especially by Jagerman in [Jag66], for the case of bandlimited functions f when N c = N b = N. In [AK] two of us extended Jagerman s result to the setting of translation invariant sampling spaces. It is natural to examine further the local error for non-symmetric intervals as in (.). For perspective, we note the close relation of this material with the uncertainty principle, see [BHW95] and [BF94]. In fact, some forms of the uncertainty principle can be viewed in terms of an interplay between localization on a time space and localization on a corresponding frequency space. In a work such as this, dealing with MRAs, frequency localization can be considered as a proection on a wavelet space, cf., the Bell Labs inequality for wavelet spaces [KP99]. Moreover, on certain sampling spaces we can easily examine local effects

3 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS 3 of the uncertainty principle, e.g., the Gibbs phenomenon on wavelet and sampling spaces [AK99], [Kar98]. In this context and as far as we know, it seems that there is an absence of systematic work dealing with the effects of the uncertainty principle on sampling spaces; we hope to address this issue on another occasion..3. Notation. We shall use the standard notation from harmonic analysis as found in [Kat68]. We also incorporate the following notation. Z is the additive group of integers and T d = R d /Z d is the d-dimensional torus. A(T d ) is the Banach algebra of absolutely convergent Fourier series Ψ(γ) = m Z d a m e πi m,γ with norm Ψ A(T d ) = m Z d a m. If Ψ A(T d ) then the Fourier coefficients a m are of the form Ψ [m] = Ψ(γ)e πi m,γ dγ. T d C(T d ) is the space of continuous functions on T d, i.e., the -periodic functions on R d. Clearly, A(T d ) C(T d ). ONB denotes orthonormal basis. Finally, a m = (A m ), m, where A m >, is the classical convention to indicate that C > such that m Z, a m CA m.. The sampling function of an MRA The following concept is fundamental in wavelet theory, e.g., see [Dau9], [Mal98], [Mey9]. Definition.. a. A dyadic multiresolution analysis (MRA) of L (R d ) is a sequence {V m : m Z} of closed linear subspaces of L (R d ) with the following properties: i. (Inclusion) m Z, V m V m+, ii. (Separation) m V m = {}, iii. (Density) m V m = L (R d ), iv. (Scaling) f(x) V m f( m x) V, v. (Orthonormality) ϕ V, a scaling function, such that {τ n ϕ : n Z d } is an ONB forv. b. We shall use the standard notation ϕ m,n (x) = md/ ϕ( m x n) for m Z, n Z d. In this section we shall give conditions on the scaling function ϕ of an MRA so that a sampling theorem similar to Theorem. holds. The main result is Theorem.6, and the sampling formula there asserts that any f V m can be written as (.) f = d f(n/ m )s( m. n), where the convergence is in the L (R d ) sense and where s is the unique sampling function of V, i.e., it satisfies s(n) = δ,n, n Z d, see also [Wal94], [Zay93]. Theorem.6 is best formulated in the setting of the following function space.

4 4 N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS Definition.. a. Let C, (Z d, T d ) be the subspace of bounded continuous functions f C b (R d ) with the property that f C, (Z d,t d ) = sup f(x + n) <. x T d d If f C, (Z d, T d ), then f L (R d ) and f is uniformly continuous on R d. b. Spaces of the type C, (Z d, T d ) are mixed norm spaces and they have a long history in analysis. The norm for f C, (Z d, T d ) is first global (over Z d ), then local (over T d ); and it is defined in the tradition of the Benedek-Panzone spaces [BP6]. Even earlier, Wiener, e.g., [Wie33], defined what we call Wiener amalgam spaces as a natural setting in which to prove Tauberian theorems, see [FS85] for a survey and [BBE89] for applications. The mixed norm for Wiener amalgam spaces is first local, then global. More recently than [BP6] but in order to deal with refinement equations and the construction of pre-wavelets, Micchelli [Mic9] introduced the analogue of C, (Z d, T d ) for measurable functions, cf., [JM9]. Also a generalization of C, (Z d, T d ) was introduced in [BZ97] to prove a class of Poisson summation formulas (PSFs), one of which is stated below in Theorem.5. Finally, Fischer [Fis95] used precisely C, (Z d, T d ) in her study of MRA and approximation theory. Example.3. a. The space C c (R d ) of compactly supported functions in R d is contained in C, (Z d, T d ). b. Let f be continuous on R d and assume there is ε > such that f(x) = ( x (d+ε) ), x. Then f C, (Z d, T d ). c. L p,q (Z d, T d ) is the space of measurable functions f on R d with the property that f L p,q (Z d,t d ) = q/p /q f(x + n) p dx, d T d where p, q [, ] and with the usual modification in the case p or q is. Similarly we can define C p,q (Z d, T d ). d. Clearly L, (Z d, T d ) L,q (Z d, T d ) L, (Z d, T d ) = L (R d ). Also, C, (Z d, T d ) C, (Z d, T d ) L (R d ). The following theorem is a consequence of definitions and the inequality, ( ) (.) f, τ n ϕ τ n ϕ(x) ϕ L (R d ) ϕ C, (Z d,t d ) f L (R d ), d which itself is a consequence of Hölder s inequality in the case {τ n ϕ} is an ONB for its closed linear span. Theorem.4. Let {V m } be an MRA of L (R d ) with scaling function ϕ C, (Z d, T d ). (In particular, ϕ is a uniformly continuous element of L (R d ) C, (Z d, T d ) L (R d ) L (R d ).) a. m Z, V m C b (R d ) L (R d ).

5 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS 5 b. m Z and f V m, f = d f, ϕ m,n ϕ m,n in L (R d ) and uniformly on R d, cf., the conclusion of Theorem.. c. Each V m is a reproducing kernel Hilbert space. In fact, for each x R d, the pointevaluation functional, V m C, defined by f f(x), not only satisfies the norm inequality f(x) C f L (R d ) but C is independent of x, e.g., (.). d. For each m, the reproducing kernel K m associated with part c is K m (x, y) = d ϕ m,n (x)φ m,n (y), and, hence, f V m, f(y) = f(x), K m (x, y). The validity of a uniform sampling expansion is often equivalent to a corresponding Poisson summation formula (PSF), e.g., [Ben97]. The appropriate PSF for C, (Z, T) is found in [BZ97] (Theorem 5). Theorem.5. Let {k λ : λ =,,...} A(T) be an approximate identity on T. For any f C, (Z, T), f(x + n) = lim kλ [n] f(n)e πinx, λ with convergence being uniform convergence on compact sets. The following result is well known, not too difficult to prove, and central to our interest. The construction we know goes back to Schoenberg [Sch73] from 946. Walter [Wal94] used the methods in the context of wavelets and sampling, cf., [Fis95]. Before stating the theorem, note that (.3) ϕ C, (Z d, T d ) implies Φ(γ) = m Z d ϕ(m)e πi m,γ A(T d ). We shall make use of Wiener s lemma on the inversion of absolutely convergent Fourier series, see [Ben97] (page ): if Ψ A(T d ) is non-vanishing on T d, then /Ψ A(T d ). For the case of Φ defined in (.3) by ϕ C, (Z d, T d ) we write (.4) Φ(γ) c k e πi k,γ. k Z d The nonvanishing of Φ is a nonvacuous hypothesis in what follows, as is illustrated in the case of the quadratic B spline. Theorem.6. Let {V m } be an MRA of L (R d ) with scaling function ϕ C, (Z d, T d ), and assume the corresponding Fourier series Φ A(T d ) is non-vanishing on T d. a. There is a unique function s V with the property that m Z and f V m C b (R d ) L (R d ), f = ( n ) dm/ f m s m,n d

6 6 N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS in L (R d ) and absolutely and, therefore, uniformly on R d. The function s is the sampling function of {V m }. and b. Further, s C, (Z d, T d ), (.5) k Z d, s(k) = s(x) = k Z d c k ϕ(x k) and ŝ(γ) = ϕ(γ) Φ(γ), {, if k =,, if k. c. {τ n s : n Z d } is an exact frame (Riesz basis) for V. In fact, from part b, we have inf γ T d Φ(γ) ŝ(γ + n) sup d γ T d Φ(γ). Example.7. Let ϕ C, (Z d, T d ) and let {τ n ϕ} be an ONB for its closed linear span V. a. If f V and the sampled values {f(n)} of f are known, then it is easy to compute { f, τ n ϕ } by the following method which is also used in Theorem.6. We know that f(n) = k f, τ k ϕ ϕ(n k), and, hence, f(n)e πi n,γ = Φ(γ) f, τ k ϕ e πi k,γ. d k Z d If Φ A(T d ) never vanishes, then we divide by Φ and obtain f, τ k ϕ = f(n)c k n. d Thus, as noted in [Dau9] (page 56), knowledge of sampled values f(n) accompanied by MRA algorithms lead to the computation of { f, ψ m,n }, where ψ is an MRA wavelet associated with ϕ. b. In determining finer properties of the sampling function s of {V m }, it becomes relevant to estimate the coefficients {c k } in (.4), assuming Φ A(T d ) is non-vanishing. In general, this is difficult to do without further conditions, see Theorem 3.5. On the other hand, in an elementary calculation, Wiener showed that if ϕ() > Φ A(T), then /Φ A(T), e.g., [Ben97] (Section 3.4). With this assumption, one can make some quantitative observations concerning the c k. c. This material, and subsequent results for Lipschitz spaces contained in C, (Z, T), require functions ϕ C, (Z, T) for which Φ > on T and {τ n ϕ} is an ONB, or even a Riesz basis, of its closed span. Such examples do exist, e.g., Meyer s wavelets [Mey9], [Dau9], [AK]; but more systematic constructions are required, cf., [BSW93], [HWW96].

7 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS 7 3. Estimating the decay of the sampling function Recall from Theorem.6 that, with natural hypotheses, if a scaling function ϕ is in C, (Z d, T d ), then its corresponding sampling function s C, (Z d, T d ). Further, we know that if f is a continuous function on R d satisfying the decay condition f(x) = ( x (d+ε) ), x, then f C, (Z d, T d ). In this section we let d =, and we shall prove (Theorem 3.6) that, with natural hypotheses, if a scaling function ϕ C, (Z, T) satisfies the decay condition ϕ(x) = ( x (η+a) ), x, for an integer η and α [, ), then its sampling function satisfies the same condition. We should point out that the hypothesis η is an artifact of our techniques. Definition 3.. Let α. A function F : T C satisfies a Lipschitz condition of order α if C > such that γ, λ T, F (γ) F (λ) C γ λ α. In this case we write F Lip α. We shall need the following well known properties of Lip α, whose proofs are found in [Lor44], [Kat68] (page 5), and [Bar64], (pages 7, 38, and 7). Proposition 3.. a. If F L (T) and < α <, then (3.) F [m] = ( m (+α) ), m, implies F Lip α. then b. If F C(T) and the modulus of continuity ω(f, δ) is defined as ω(f, δ) = sup F (γ + λ) F (γ), γ, λ δ (3.) F Lip β if and only if ω(f, δ) = (δ β ), δ, and (3.3) m Z\{}, F [m] ω (F, ). m Lemma 3.3. Let {u m : m Z}, {v m : m Z}, {w m : m Z} C and let µ, ν > satisfy the following conditions: {w m } l p (Z), p (, ], (3.4) u m = ( m µ ) and v m = ( m ν w m ), m, (3.5) ν µ, and (3.6) µ > p p Then (3.7) =, p (, ), resp., µ >, p =. p u v m = ( m µ ), m. Z

8 8 N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS Proof. By (3.4) assume there are C µ, C ν > such that u m C µ m µ and v m C ν m ν w m. Pick m and let λ m be the non-negative integer for which λ m m / < λ m +. Then (3.8) u v m Z C ν u w m m ν + C µ λ m µ v m, >λ m where ω C for all Z. If λ m, then m m m λ m m /; and if > λ m then µ (λ m + ) µ ( m /) µ. Thus the right side of (3.8) is bounded by ( ) ν ( ) µ C ν u w m + CC µ v m m m λ m >λ m (3.9) ( ) µ [ ] C ν {u } m lp (Z) {w } l p (Z) + CC µ {v } l (Z). Note that {v } l (Z) by (3.5) and since {w } l p (Z); and {u } l p (Z) by (3.6). (3.7) follows by combining (3.8) and (3.9). Lemma 3.4. Let η be an integer and let α [, ). Define Φ(γ) = m Z ϕ(m)e πimγ, where {ϕ(m) : m Z} C satisfies the condition (3.) ϕ(m) = ( m (η+α) ), m. Assume Φ > on T. Then Φ, /Φ A(T) for all ; and, if /Φ (γ) = m Z d(m)e πimγ, then d m = ( m (η+α)+ ), m. Proof. i. Let < α <. By Wiener s lemma on the inversion on non-vanishing absolutely convergent Fourier series, we have /Φ A(T); and since A(T) is a Banach algebra under pointwise multiplication, we have /Φ A(T) for all. Also, for fixed k η (3.) Φ (k) (γ) = m Z d b (k) m e πimγ A(T), where b (k) m = ( πim) k ϕ(m), since (3.) m, b (k) m K m (η+α)+k K m α. In fact, (3.) is a consequence of (3.) and the fact that the derived series converges uniformly, cf., [Ben97] (page 8).

9 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS 9 ii. With < α <, Lorentz result (Proposition 3.a) coupled with (3.) allow us to conclude that Φ (k) Lip α for k η. Also note that products of Lip α functions on T are Lip α, and that, (Φ ) () = Φ (+) Φ () A(T). Further, the product of F Lip α and Φ is also in Lip α on T. This is a consequence of the fact that α and that (Φ ) () L (T), which allows us to find a uniform constant K in the following calculation: (F Φ )(γ) (F Φ )(λ) C γ λ α Φ L (T) + K γ λ F L (T) C γ λ α. iii. We now write d m, m, as d m = = (πim) Φ(γ) e πimγ dγ = πim (3.3) = ( )η (πim) η Φ(γ) 3 Φ () (γ)e πimγ dγ, [ ] 3Φ(γ) 4 Φ () (γ) + Φ(γ) 3 Φ () (γ) e πimγ dγ Ψ(γ)e πimγ dγ, where Ψ Lip α by the discussion in part ii. Since Ψ Lip α, we invoke Proposition 3., viz., (3.) and (3.3), in conunction with (3.3) to assert that d m C m (η ) ω(ψ, ( m ) α ) = ( m (η ) α ), m. This is the desired result for η and < α <. iv. Now let α =. In this case we have d m = C (πim) η Ψ (γ)e πimγ dγ, where Ψ A(T) by the discussion in part ii. We observe that Φ (η ) is in L (T), noting that b m (η ) = ( m ) as m. Thus, it is easy to see that the derivative of Ψ exists and is in L (T), and that d m = C (πim) η Thus, d m = ( m (η ) ) as m. Ψ () (γ)eπimγ dγ. Theorem 3.5. Let η be an integer and let α [, ). Define Φ(γ) = m Z ϕ(m)e πimγ, where {ϕ(m) : m Z} C satisfies condition (3.). Assume Φ > on T and set as in (.4). Then m Z, c m = Φ(γ) eπimγ dγ (3.4) c m = ( m (η+α) ), m.

10 N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS Proof. i. First we consider the case < α <. Clearly Φ A(T) by (3.), and hence /Φ A(T) since Φ > on T. Using the notation from Lemma 3.4 we have Φ () (γ) = m Z b () m e πimγ A(T), where b () m = πimϕ(m) K m (η+α)+, m. For m we compute c m = πim Φ(γ) Φ () (γ)e πimγ dγ. Thus, by Parseval s theorem, e.g., [Ben97] (Theorem 3.4.), we have (3.5) c m = πim Z where {d m } is defined in Lemma 3.4, and where b () d m, (3.6) b () = ( (η+α)+ ) and d = ( (η+α)+ ),, by Lemma 3.4 and the aforementioned bound on {b () }. We now apply Lemma 3.3 directly to the data in (3.6): µ = η + α, ν = η + α, p =, w =, u = b (), and v = d for each. Then µ = ν + a > and so (by Lemma 3.3) Z b () d m = ( m (η+α)+ ), m. Combining this estimate with (3.5) we obtain the desired result. ii. Now let η and α =. The case η > is an immediate consequence of part i by setting α =, so only the case η = and α = remains to be checked. For this case we follow the proof in part i and obtain (3.5), i.e., c m = πim where b () = ( ),, and d = Φ(γ) e πiγ dγ = π Z b () d m, Φ(γ) 3 Φ () (γ)e πiγ dγ = π δ,. Clearly, the δ designate the Fourier coefficients of the function Φ () Φ 3, which, in turn, is an element of L (T). This latter assertion is a consequence of the facts that Φ 3 C(T) and that the Fourier coefficients b () of Φ () are dominated by,. We can now apply Lemma 3.3 by setting µ =, ν =, p =, w = δ, u = b (), and v = d for each ; and we obtain Z b () d m = ( m ), m. Combining this with (3.5) we obtain c m = ( m ), m.

11 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS Theorem 3.6. Let η be an integer and let α [, ). Further, let {V m } be an MRA of L (R) with scaling function ϕ C, (Z, T) satisfying the condition (3.7) ϕ(x) = ( x (η+α) ), x. Assume the corresponding Fourier series Φ A(T), with Fourier coefficients {ϕ(m)}, is non-vanishing on T. Then the sampling function s of {V m } satisfies the condition (3.8) s(x) = ( x (η+α) ), x. Proof. From Theorem.6 we know that s(x) = m Z c mϕ(x m), and because of (3.7) and Theorem 3.5, we know that c m = ( m (η+α) ), m. For any x R\{}, let m x N {} satisfy m x x / < m x +. Then (3.9) s(x) = c m ϕ(x m) + c m ϕ(x m) = s (x) + s (x), m m x m >m x where s, resp., s, is the first, resp., second, sum in (3.9). If m m x, then and so x m x m = x m x m x x /, (3.) s (x) C ϕ ( x ) (η+α) m m x c m, where C ϕ is a bound from (3.7) independent of x. Further, because of Theorem 3.5 we also have the estimate (3.) s (x) C c (m x + ) ( ) x (η+α) (η+α) ϕ(x m) < C c sup ϕ(t m), m Z t R where C c is a bound from (3.4) (Theorem 3.5) independent of x. The sup term in (3.) is the C, (Z, T) norm of ϕ. Combining (3.) and (3.), we obtain (3.8) with a bound C s = η+α ( C ϕ /Φ A(T) + C c ϕ C, (Z,T)). m Z The bound C s at the end of the proof of Theorem 3.6 grows with η, and this is meaningful in that it reveals the trade-off between the bound implicit in (3.8) and the rate of decay asserted by (3.8). The trade-off can be made more technically explicit with the goal of applying Theorem 4. (see below), even when φ has compact support and, therefore, s has fast decay. 4. A local error for wavelet expansions In this section we shall estimate the local error for wavelet expansions.

12 N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS Definition 4.. Let N, N Z for N < N, let h = m for m Z, and let d =. The local error of the sampling formula (.), i.e., f = ( n ) m/ f m s m,n, is (4.) E N,N f(x) f(x) where x [N h, N h]. N n=n f(nh)s(h x n), We consider an arbitrary interval I R, and let a = inf{x I}, b = sup{x I}, and (4.) M a = sup{n Z : nh a} and M b = inf{n Z : nh b}. We define (4.3) P a = M a N a and P b = N b M b, where N a < M a and N b > M b are integers. Notationally, the left side of (.) can be written as Nb sup E Na,N b f(x) = sup x I x I f(x) f(n/ m )s( m x n). n=n a The following result does not depend on the results of Section 3, nor does it depend essentially on the MRA setup except to determine a sampling function; but it does depend significantly on the methods in [HT6], [Jag66]. Theorem 4.. Let {V m } be an MRA of L (R) with scaling function ϕ C, (Z, T). Assume the corresponding Fourier series Φ A(T) is non-vanishing on T, and let s V C b (R) L (R) be the corresponding uniquely determined sampling function, see Theorem.6. Let I R be a bounded interval with associated constants a, b, M a, M b as in (4.), where h = m, and let P a, P b be as in (4.3) for N a < M a and N b > M b. Assume (4.4) s(x) = ( x β ), x for some fixed β > /, and set A N (h x) = ( n>n n h x β ). For each m Z and f V m define ( K N = K N (f) = h ) ( f(nh) and L N = L N (f) = h n>n n<n f(nh) ). a. There is a constant C, arising from (4.4) and independent of f V m, such that x [N a h, N b h], E Na,N b f(x) Ch [ KNb (f)a Nb (h x) + L Na (f)a Na ( h x) ], noting I [M a h, M b h] [N a h, N b h].

13 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS 3 b. Let C be the constant of part a. Then [ ] sup E Na,N b f(x) C hβ K Nb (f) x I β (P b h + r b ) β / + L Na (f) (P a h + r a ) β /, where r a = a M a h h and r b = M b h b h are non-negative. Proof. a. If x = N a h or x = N b h then the local error (4.) is. For x (N a h, N b h) we have E Na,N b f(x) C f(nh) (n h x) β + f(nh) (h x n) β n>n b n<n a by (4.), where C is the constant arising from (4.4). Thus, part a is a consequence of the Cauchy-Schwarz inequality, since, for example, if x N b h and n > N b then n xh >. b. We first note that if x I then ( A Nb (h x) N b (y h dy x) β ) = h β / β (N b h x) β /, with a similar estimate for A Na ( h x). Combining this with part a, we have [ ] (4.5) E Na,N b f(x) Chβ K Nb (f) β (N b h x) β / + L Na (f) (x N a h) β / if x I. Now, note that if x I then and N b h x = N b h M b h + M b h x P b h + M b h b = P b h + r b x N a h r a + P a h. The function g N,h,β (x) = (Nh x) β+/ is strictly increasing for x < Nh, and g N,h,β ( x) is strictly decreasing for x > Nh. Thus, for a x b, we have and g Nb,h,β(x) g Nb,h,β(b) = g Pb,h,β( r b ) g Na,h,β(x) g Na,h,β( a) = g Pa,h,β( r a ). Substituting these inequalities into (4.5) yields part b. Corollary 4.3. Assume the setup, hypotheses, and notation of Theorem 4.. Let P = P a = P b, and, for each m, let K m be the reproducing kernel of V m, e.g., Theorem.4. Then ( ) C sup E Na,N b f(x) x I β P β / f(nh) ( ) C β P β / K m (nh, nh) f L (R).

14 4 N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS Proof. The first inequality is elementary from Theorem 4.b using the same value of C as in Theorem 4.. The second inequality is a consequence of the definition of the reproducing kernel: f(nh) = f(x), K m (x, nh) f L (R) K m (x, nh), K m (x, nh) = f L (R) K m (nh, nh). We now invoke Theorem 3.6 in order to obtain the conclusion of Theorem 4., but, because of the MRA setting, to obtain this conclusion by only making a decay hypothesis on the scaling function ϕ (and not on its associated sampling function s). The proof of Theorem 4.4 is immediate from Theorems 3.6 and 4.. Theorem 4.4. Assume the setup, hypotheses, and notation of Theorem 4. exclusive of the assumption (4.4) on the decay of the sampling function. Let η and α [, ), and assume ϕ(x) = ( x (η+α) ), x. There is C > such that h (η+α) [ E Na,N b f(x) C (η + α) f V m, sup x I K Nb (f) (P b h + r b ) (η+α) / + L Na (f) (P a h + r a ) (η+α) / 5. A sampling theorem based on the Gabor transform In this section we give a sampling theorem for bandlimited functions, using the Gabor transform for L (R). For more details see [BF94], as well as [Ben89] for the L (R) case. For other fully developed aspects of the Gabor theory see [FS98], [Grö]. Definition 5.. a. Let g L (R) L (R). The Gabor transform G g of f L (R) is defined to be (G g f) x (ω) = f(t), g(t x)e πitω = f(t)g(t x)e πitω dt. b. If G g f is the Gabor transform of f L (R), then the inverse Gabor transform of f is given by (5.) ((G g f) x (ω)) (t) = (G g f) x (ω)g(t x)e πitω dxdω, g L (R) and f(t) = ((G g f) x (ω)) (t) a.e. on R. Theorem 5.. Let f P W Ω and g P W Γ. If P (y) g(y + x)g(x)dx is the positive definite autocorrelation of g, then ( n ) ( (5.) f(t) = Λ g f P t n ) sin[π(t n/(λ))], a.e., Λ Λ π(t n/(λ)) where Λ = Γ + Ω. ].

15 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS 5 Proof. For a fixed x R the Gabor transform (G g f) x (ω) may be considered as the Fourier transform of f(t)g(t x). Using Parseval s formula, we have (G g f) x (ω) = e πixω γ Ω f(γ)ĝ(ω γ)e πixγ dγe πixω. Obviously, f(t)g(t x) P W Λ, where Λ = Γ + Ω. Thus, the Fourier transform (G g f) x (ω) may be developed in a Fourier series, as follows: (G g f) x (ω) = Λ ( (G g f x )(γ)e iπnγ/λ dγ γ Λ ) e iπnω/λ = Λ f( n Λ )g( n Λ x)eπinω/λ, where ω [ Λ, Λ). The second equality follows from the definition of G g f for fixed x and the Fourier inversion theorem. Each series converges a.e. on R. After applying Fubini s theorem, we substitute the second series into (5.) to obtain (5.3) f(t) = = Λ g L (R) g L (R) g(t x) Λ Λ f( n Λ )g( n x)g(t x) Λ (G g f) x (ω)e πitω dωdx ω Λ e π(t n/(λ))ω dωdx, a.e. Interchanging summation and integration in the last term of (5.3), noting that term by term integration of Fourier series of integrable functions is allowed whether they converge or not, we see that (5.3) becomes f(t) = Λ g L (R) and this is the desired result. f( n n )P (t Λ Λ n/(λ))] )sin[π(t, a.e., π(t n/(λ)) Corollary 5.3. Let K(t, x) = g(t x) be the (sinc) reproducing kernel of the space P W Γ, where (x, t) R. If Λ = Γ + Ω, then f P W Ω, f(t) = Λ g f( n Λ )g(t n n/(λ))] )sin[πλ(t, a.e. Λ π(t n/(λ)) L (R) Proof. If h P W Γ, then h(x) = h( ), K(, x) = h( ), g( x) = h(y)g(y x)dy. Thus, if P is defined as in Theorem 5., then P (t n/(λ)) = g(t n/(λ)), and the result follows from (5.). Corollary 5.4. Assume the setting and notation of Theorems 4. and 5.. Let s(t) = P (t)sin(πtλ) πtλ g L (R) t,

16 6 N. ATREAS, J. J. BENEDETTO, AND C. KARANIKAS where P (y) = g(y + x)g(x)dx, and suppose P (y) = ( y β ), y. If h = (Λ), then f P W Ω, sup E Na,N b f(x) C hβ x I [ K Nb (f) β (P b h + r b ) β / + L Na (f) (P a h + r a ) β / Proof. P W Ω can be viewed as a V. The decay hypothesis on P implies (4.4); and the definition of s yields the sampling formula f = f(n/(λ))τ n/λ s by Theorem 5.. This formula, coupled with the proof of Theorem 4., gives the result. ACKNOWLEDGEMENTS The authors would like to thank Alexander Powell for indispensible input to the final presentation of this work. We also appreciate the referee s thoughtful report, and we have incorporated all of his suggestions. (The referee referred to himself.) ]. References [AK99] N. Atreas and C. Karanikas, Gibbs phenomenon on sampling series based on Shannon s and Meyer s wavelet analysis, J. Fourier Analysis and Applications 5 (999), [AK], Truncation error on wavelet sampling expansions, J. Computational Analysis and Appl. 5 (), [Bar64] N. K. Bary, A Treatise on Trigonometric Series, Volume I, The MacMillan Company, New York, 964. [BBE89] J. J. Benedetto, G. Benke, and W. R. Evans, An n-dimensional Wiener-Plancherel formula, Advances in Applied Math. (989), [Ben89] J. J. Benedetto, Gabor representations and wavelets, Contemporary Mathematics 9 (989), 9 7. [Ben97], Harmonic Analysis and Applications, CRC Press, Boca Raton, FL, 997. [BF94] J. J. Benedetto and M. W. Frazier (eds.), Wavelets: Mathematics and Applications, CRC Press, Boca Raton, FL, 994. [BF] J. J. Benedetto and P. J. S. G. Ferreira (eds.), Modern Sampling Theory: Mathematics and [BHW95] Applications, Birkhäuser, Boston,. J. J. Benedetto, C. Heil, and D. F. Walnut, Differentiation and the Balian-Low theorem, J. Fourier Analysis and Applications (995), [BP6] A. Benedek and R. Panzone, The spaces L p, with mixed norm, Duke Math. J. 8 (96), [BSW93] A. Bonami, F. Soria, and G. Weiss, Band-limited wavelets, J. Geometric Analysis 3 (993), [BZ97] [BZ3] [Dau9] J. J. Benedetto and G. Zimmermann, Sampling multipliers and the Poisson summation formula, J. Fourier Analysis and Applications 3 (997), J. J. Benedetto and A. I. Zayed (eds.), Sampling, Wavelets, and Tomography, Birkhäuser, Boston, 3. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Math. SIAM, Philadelphia, 99. [Fis95] A. Fischer, Multiresolution analysis and multivariate approximation of smooth signals in C B(R d ), J. Fourier Analysis and Applications (995), 6 8. [FS85] J. J. Fournier and J. Stewart, Amalgams of L p and l q, Bull. Amer. Math. Soc. 3 (985),. [FS98] H. G. Feichtinger and T. Strohmer (eds.), Gabor Analysis and Algorithms: Theory and Applications, Birkäuser, Boston, 998. [Grö] K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston,. [HT6] H. D. Helms and J. B. Thomas, Truncation error of sampling-theorem expansions, Proc. IRE 5 (96), [HWW96] E. Hernández, X. Wang, and G. Weiss, Smoothing minimally supported frequency wavelets, Part I, J. Fourier Analysis and Applications (996),

17 LOCAL SAMPLING FOR REGULAR WAVELET AND GABOR EXPANSIONS 7 [Jag66] D. Jagerman, Bounds for truncation error of the sampling expansion, SIAM J. Applied Math. 4 (966), [JM9] R.-Q. Jia and C. A. Micchelli, Using the refinement equations for the construction of pre-wavelets ii: powers of two, Curves and Surfaces (P. J. Laurent, A. LeMéhauté, and L. L. Schumaker, eds.), Academic Press, Boston, 99, pp [Kar98] C. Karanikas, Gibbs phenomenon in wavelet analysis, Results Math. 34 (998), [Kat68] Y. Katznelson, An Introduction to Harmonic Analysis, Dover Publications, New York, 968. [KP99] C. Karanikas and G. Proios, The Slepian-Pollak energy inequality for wavelet and frames, Appl. Anal. 7 (999), [Log84] B. F. Logan, Jr., Signals designed for recovery after clipping. III. Generalizations, AT&T Bell Labs. Tech. J. 63 (984), no. 3, [Lor44] G. G. Lorentz, Fourier-koeffizienten und funktionenklassen, Math. Zeit. 5 (944), [Mal98] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Boston, 998. [Mar] F. Marvasti (ed.), Nonuniform Sampling: Theory and Practice, Plenum / Kluwer, New York,. [Mey9] Y. Meyer, Ondelettes et Opérateurs, Hermann, Paris, 99. [Mic9] C. A. Micchelli, Using the refinement equation for the construction of pre-wavelets, Numerical Algorithms (99), [Sch46] I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quarterly J. Applied Math. 4 (946), and 4. [Sch73], Cardinal Interpolation, CBMS-NSF Series in Applied Math. SIAM, Philadelphia, 973. [Wal94] G. G. Walter, Wavelets and other Orthogonal Systems, CRC Press, Boca Raton, FL, 994. [Wie33] N. Wiener, The Fourier Integral and Certain of its Applications, MIT Press, Cambridge, MA, 933. [Zay93] A. I. Zayed, Advances in Shannon s Sampling Theory, CRC Press, Boca Raton, FL, 993. Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, Greece address: natreas@auth.gr Department of Mathematics, University of Maryland, College Park, MD 74, USA address: b@math.umd.edu URL: b Department of Informatics, Aristotle University of Thessaloniki, Thessaloniki, Greece 546 address: karanika@ccf.auth.gr