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Recovering functions from the modulation space 5 FW 1 0 2 Jeff Ledford n a J Abstract 0 1 In this short note we show that functions in the modulation space ] FW = {f : Pj∈Znkfˆ(·+2πj)kL∞([−π,π]n) < ∞} enjoy similar recovery A properties as band-limited functions. If {φα} is a regular family of cardinal interpolators, then one can build an approximand of f using the C fundamentalfunctionscorrespondingtoφ . Thentakingtheappropriate α . h limit, one recovers f both in norm and pointwise. t a m 1 Introduction [ This note continues the study ofapproximationandrecoveryoffunctions using 1 v cardinalinterpolants. Thisproblemhasarichhistorywhichhastendedtofocus 2 on a specific approximation scheme, e.g. spline, multiquadric. A few results in 1 this area may be found in [7], [8], and [1]. Each deals with the recovery of 3 band-limited functions. 2 It was the goal of [5] to unite these methods under a single framework, 0 . regular families of cardinal interpolators. The goal of this work is to use the 1 same framework and move beyond band-limited functions. A similar study has 0 5 been done for scattered data in [6], although in a limited context. The main 1 resultthereisanalogoustoTheorem2below. Similarworkhasalsobeencarried : out by Hamm in [3], where Sobolev functions are recovered. v i The remainder of this note is organized as follows. Section 2 contains defi- X nitions and basic facts needed to set up our problem. The main result is stated r and proved in Section 3, while the final section contains examples of regular a families of cardinal intepolators. 2 Definitions and Basic Facts We begin with a convention for the Fourier transform. Definition 1. The Fourier transform of a function f(x) ∈ L1(Rn), is defined to be the function fˆ(ξ):=(2π)−n/2 f(x)e−ihx,ξidx. Z Rn 1 We make the usual extension to the class of tempered distributions using this convention. We will be interested in the following modulation spaces. Let 1 ≤ p ≤ ∞ and define W(L ,ℓ ):=f ∈L (R): kf(·+2πj)k <∞, and p 1  ∞ jX∈Zn Lp([−π,π]n)  FW(L ,ℓ ):=f :fˆ∈W(L ,ℓ )  p 1 p 1 n o The space W(L ,ℓ ) is called the Wiener space. These spaces are sometimes ∞ 1 called Wiener amalgam spaces. To avoid cumbersome notation we will abbre- viate these spaces as W and FW , respectively. Additionally, we will write p p W := W(L ,ℓ ) for the Wiener space. For more information on modulation ∞ 1 spaces the reader may consult Capters 6 and 11 of [2]. We will also make use of the classical Paley-Wiener space PW, given by PW :={f ∈L (R):fˆ(ξ)=0 a.e ξ ∈/ [−π,π]n}. 2 The norms on these spaces are given by kfk := kf(·+2πj)k , Wp Lp([−π,π]n) jX∈Zn kfkFWp := kfˆ(·+2πj)kLp([−π,π]n), and jX∈Zn kfk :=kfˆk PW L2([−π,π]n) Note the straightforward inclusion FW ⊂ FL (Rn) and FW ⊂ FW for 1 ≤ p p p p≤∞. Our next definitions come from [5]. Definition 2. We say that a function φ is a cardinal interpolator if it satisfies the following conditions: (H1) φ is a real valued slowly increasing function on Rn, (H2) φˆ(ξ)≥0 and φˆ(ξ)≥δ >0 in [−π,π]n, (H3) φˆ∈Cn+1(Rn\{0}), (H4) There exists ǫ>0 such that if |α|≤n+1, Dαφˆ(ξ)=O(kξk−(n+ǫ)) as kξk→∞, (H5) For any multi-index α, with |α|≤n+1, |α| Dαjφˆ |α| Y j=1 ∈L∞([−π,π]n) where α =α. [φˆ]|α|+1 X j j=1 2 Definition3. Wecallafamilyoffunctions{φ } aregularfamilyofcardinal α α∈A interpolators if for each α ∈ A ⊂ (0,∞), φ is a cardinal interpolator and in α addition to this, we have: φˆ (ξ+2πj) (R1) For j ∈ Zn \{0}, ξ ∈ [−π,π]n, define M (ξ) = α , then for j,α φˆ (ξ) α j ∈Zn\{0}, lim M (ξ)=0 for almost every ξ ∈[−π,π]n. j,α α→∞ (R2) There exists M ∈ l1(Zn \{0}), independent of α, such that for all j ∈ j Zn\{0}, M (ξ)≤M . j,α j These definitions allow us to form a fundamental function L associated to α φ , defined by its Fourier transform α φˆ (ξ) Lˆ (ξ)=(2π)−n/2 α . α φˆ (ξ+2πj) j∈Zn α P We will need a few properties which may be found in [5]. Proposition 1 (Proposition 3, [5]). If {φ } is regular a regular family of car- α dinal interpolators, then lim Lˆ (ξ+2πj)=0 for a.e. ξ ∈[−π,π]n and in α→∞X φα j6=0 L1([−π,π]n). Proposition 2 (Theorem 1, [5]). Suppose that {φ } is a regular family of α cardinal interpolators and let f ∈PW . Then we have the following limits: π (a) lim f(j)L (·−j)−f =0, α→∞(cid:13)(cid:13)jX∈Zn φα (cid:13)(cid:13)L2(Rn) (cid:13) (cid:13) (cid:13) (cid:13) (b) lim f(j)L (·−j)−f =0 uniformly in Rn. α→∞(cid:12)(cid:12)jX∈Zn φα (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Ourgoalistobuildapproximandsofagivenfunctionf withsufficientlynice properties. To do this we first need {f : k ∈ Zn}. We define f by its Fourier k k transform fˆ(ξ)=fˆ(ξ+2πk)χ (ξ). (1) k [−π,π]n Now our approximandwill take the form J [f](x):= f (j)L (x−j)e2πihx,ki. (2) α k α j,Xk∈Zn 3 Main Result Our main result is the following theorem. 3 Theorem 1. Suppose that f ∈ FW and {φ } a regular family of cardinal α interpolators. We have αl→im∞kf −Jα[f]kFWp =0, for 1≤p≤∞. Proof. We note that f ∈ FW implies that f ∈ FW for all 1 ≤ p ≤ ∞, p additionally {f :k ∈Zn}⊂PW. As a result, we may use the formula k (2π)n/2fˆ(ξ)= f (j)e−ihj,ξi, a.e. ξ ∈[−π,π]n. (3) k k jX∈Zn Now we may begin our calculation. kf −Jα[f]kFWp = kfˆ(·+2πl)−J[[f](·+2πl)k α Lp([−π,π]n) lX∈Zn = kfˆ(·+2πl)−(2π)n/2 Lˆ (·+2π(l−k))fˆk α k Lp([−π,π]n lX∈Zn kX∈Zn ≤ k(1−(2π)n/2Lˆ )fˆk α l Lp([−π,π]n lX∈Zn +(2π)n/2 k Lˆ (·+2π(l−k))fˆk α k Lp([π,π]n) lX∈Zn Xk6=l =(2π)n/2 k Lˆ (·+2πk)fˆk α l Lp([−π,π]n lX∈Zn Xk6=0 +(2π)n/2 k Lˆ (·+2πk)fˆ k α l−k Lp([π,π]n) lX∈Zn Xk6=0 ≤2(2π)n/2 M kfˆk k l Lp([−π,π]n) Xk6=0 lX∈Zn =2(2π)n/2 MkkfkFWp X k6=0   The firsttwo equalitiesare justthe definitionandthe formula(3). The first inequalityisthetriangleinequality. Thenextequalityfollowsfromthefactthat 1−Lˆ (ξ)=(2π)n/2 Lˆ (ξ+2πk). α α X k6=0 The final inequality is (R2), where we have switched the sums with Tonelli’s theorem. Now Proposition 1 and the Dominated Convergence theorem finish the proof. 4 This theorem leads to a pointwise result. Corollary 1. Under the hypotheses of Theorem 1, we have lim |f(x)−J [f](x)|=0 α α→0 uniformly on Rn. Proof. To see this, we use the Fourier integral representation. |f(x)−J [f](x)| α ≤(2π)−n/2 |fˆ(ξ)−J[[f](ξ)|dξ Z α Rn =(2π)−n/2 kfˆ(·+2πl)−J[[f](·+2πl)k α L1([−π,π]n) lX∈Zn ≤(2π)1−1/p−n/2 kfˆ(·+2πl)−J[[f](·+2πl)k α Lp([−π,π]n) lX∈Zn =(2π)1−1/p−n/2kf −Jα[f]kFWp Herewehaveperiodizedthefirstintegral,thenappliedHölder’sinequality. Now taking the limit yields the result. We also get an L result of sorts. p Corollary 2. Under the hypotheses of Theorem 1, we have αli→m0kf −Jα[f]kFLp(Rn) =0. Proof. To see this we need only note that kfkF(Rn) =kfˆkLp(Rn) ≤ kfˆ(·−2πk)kLp([−π,π]n) =kfkFWp. kX∈Zn For certain p, we can relax the conditions on f somewhat. Specifically, we have the following. Theorem 2. Suppose that f ∈ FW and {φ } is a regular family of cardinal 2 α interpolators, then we have (a) lim kf −Jα[f]kFW2 =0, α→∞ (b) αl→im∞kf −Jα[f]kL2(Rn) =0, and (c) lim |f(x)−J [f](x)|=0, uniformly on Rn. α α→∞ Proof. The proof is the same mutatis mutandis. We simply note that in this case we can use (3) and carry out the calculation as before. 5 4 Examples In this section we provide concrete examples of families of functions which ex- hibitthisbehavior. Sinceanyfamilyofregularcardinalinterpolatorswillwork, we can use the examples found in [5]. For more information regarding these functions the reader may consult [5] and the references found there. 4.1 Polyharmonic Cardinal Splines Suppose that ∆ is the n-dimensional Laplacian and ∆kf = ∆(∆k−1f). For k ≥1,if∆kf =0onRn\Znandf ∈C2k−2(Rn),thenf iscalledapolyharmonic spline. We note that in this case we have φˆ (ξ) = kξk−2k. The appropriate k family of cardinal interpolators is given by {φ :k ∈N}. k 4.2 Gaussians The family given by {e−k·k2/(4α) : α ≥ 1} is a regular family of cardinal interpolators. 4.3 Multiquadrics The family given by {(k·k2+c2)α :j ∈N,c>0 fixed} where {α }⊂[1/2,∞) j j and dist({a },N) > 0 is a regular family of cardinal interpolators. This is also j true if we consider {(k·k2+c2)α : c ≥ 1} here we may take α ∈ R\N , this 0 final result was shown in [4], with a partial result appearing in [5]. References [1] B.Baxter,“Theasymptoticcardinalfunctionofthe multiquadraticϕ(r)= (r2+c2)1/2 as c→∞,” Comput. Math. Appl. 24 (1992), no. 12, 1-6 [2] K. Gröchenig, Foundations of Time Frequency Analysis, Birkhäuser, Boston, MA, 2001 [3] K.Hamm,“ApproximationRatesforInterpolationofSobolevFunctionsvia Gaussians and Allied Functions,” J. Approx. Theory, 189, (2015), 101-122 [4] K. Hamm J. Ledford, “Cardinal interpolationwith generalmultiquadrics,” (preprint) arXiv:1501.01899 [5] J. Ledford, “On the convergence of regular families of cardinal interpolators," Adv. Comp. Math. (to appear) [6] “Recovering functions from the Paley-Wiener amalgam space,” (preprint) arXiv:1311.5169 [7] W.MadychS.Nelson,“PolyharmonicCardinalSplines,” J.Approx.Theory 60, 1990), no.2, 141-156 6 [8] S.RiemenschneiderandN.Sivakumar,“OncardinalinterpolationbyGaus- sian radial-basis functions: properties of fundamental functions and esti- mates for Lebesgue constants” J. Anal. Math. 79, (1999), 33-61. 7