On the Structure and Interpolation Properties of Quasi Shift-invariant Spaces
aa r X i v : . [ m a t h . F A ] F e b ON THE STRUCTURE AND INTERPOLATION PROPERTIESOF QUASI SHIFT-INVARIANT SPACES
KEATON HAMM AND JEFF LEDFORD
Abstract.
The structure of certain types of quasi shift-invariant spaces, whichtake the form V ( ψ, X ) := span L { ψ ( · − x j ) : j ∈ Z } for an infinite discreteset X = ( x j ) ⊂ R is investigated. Additionally, the relation is explored be-tween pairs ( ψ, X ) and ( φ, Y ) such that interpolation of functions in V ( ψ, X )via interpolants in V ( φ, Y ) solely from the samples of the original function ispossible and stable. Some conditions are given for which the sampling problemis stable, and for which recovery of functions from their interpolants from afamily of spaces V ( φ α , Y ) is possible. Introduction
At the heart of modern signal processing and sampling theory are two funda-mental questions: when is sampling a class of functions at a given point-set stable?And if the sampling is stable, how may functions in the given class be reconstructedfrom their samples? While these questions may be based around applications andare indeed of import in engineering disciplines, they lead quite quickly to some deeptheoretical and structural mathematical problems. The origin of classical samplingtheory lies in the observation that functions in the Paley–Wiener space of bandlim-ited functions in L ( R ) whose Fourier transforms are supported in the torus T canbe recovered both in L and uniformly on R by a cardinal sine series: f ( x ) = X j ∈ Z f ( j ) sinc( x − j ) . This is equivalent to the observation that the exponential system ( e − ij · ) j ∈ Z is anorthonormal basis for L ( T ).However, much literature in the modern era has been devoted to breaking awayfrom the assumption that functions are bandlimited (which implies, in particular,that the functions are analytic). Consequently, many interesting function spaceshave been discussed – Wiener amalgam spaces, modulation spaces, spaces of fi-nite rate of innovation, shift-invariant spaces, Sobolev spaces, and Triebel–Lizorkinspaces to name a few.In particular, shift-invariant spaces have been popularized in many areas, in-cluding functional analysis, harmonic analysis, and approximation theory (in thelatter field, they have been well-studied specifically in the setting of radial basisfunction approximations [10, 11, 12, 17, 38, 39, 44]). A shift-invariant space hasthe form V ( ψ ) := span { ψ ( · − j ) : j ∈ Z } , where the closure is taken in L p ( R ) forsome p ∈ [1 , ∞ ] (see, for example, [13, 37]). As a matter of terminology, we call ψ Mathematics Subject Classification.
Key words and phrases.
Shift-invariant Space, Quasi Shift-invariant Space, Nonuniform Sam-pling, Interpolation. the generator of the space (note it is sometimes called the window or kernel). Theobjects of study in this paper are quasi shift-invariant spaces V ( ψ, X ) := span { ψ ( · − x j ) : j ∈ Z } , where the closure is taken in L . Quasi shift-invariant spaces in this form with X being infinite have been considered in [5, 26], and are also implicitly considered in[62]. Recently, stability results for Riesz bases in such spaces were considered in[20].The purpose of this article is twofold: to determine how the structure of quasishift-invariant spaces compares to the structure of shift-invariant spaces under thesame assumptions on the generators; and secondly, to discuss the recovery of func-tions in quasi shift-invariant spaces via their interpolants from other quasi shift-invariant spaces. Motivated by the study of shift-invariant spaces, we consider thefollowing problems. Problem 1.
Under standard assumptions on the generators ψ , and discrete trans-lation sequences X ⊂ R , what is the structure of V ( ψ, X )? In particular, is thespace a closed subspace of L ? Do the translates of ψ form a natural basis for thespace? Problem 2. a) Under what conditions on ψ, φ, X , and Y is interpolation of func-tions in V ( ψ, X ) via interpolants in V ( φ, Y ) possible, and uniquely determined?b) Under what conditions may f ∈ V ( ψ, X ) be recovered from its interpolants ina family of quasi shift-invariant spaces ( V ( φ α , Y )) α ∈ A ?Answers to the first problem in the shift-invariant case (i.e. X = Z ) are drivenby the fact that V (sinc , Z ) = P W π , which of course is a closed subspace of L ( R ),and the integer translates of sinc form an orthonormal basis for P W π . The sec-ond problem is one of scattered-data interpolation and has been studied by severalauthors. For instance, Dyn and Michelli [21] provide results for finite multivari-ate interpolation using conditionally positive definite functions, while Jetter andSt¨ockler [36] examine irregular sampling where one of the spaces is a shift-invariantspline space. Similar considerations, including reconstruction algorithms may befound in [2, 3]. More recently, Atreas [5] considers spaces V ( ψ, X ) under essentiallythe same assumptions made below ((A1) and (A2) in Section 3) and gives a re-construction formula in the spirit of the classical Whittaker–Kotel’nikov–Shannon(WKS) sampling theorem; some results in this article may be viewed as approxi-mate reconstruction methods in the same vein. Radial basis functions (RBFs) havebeen employed to attack this problem as well, [52, 53]. The literature on RBFinterpolation is vast, and the reader is encouraged to consult [15] and [64] for amore general discussion of this problem.These works, and the results of [45], which solve Problem 2 in the special casewhen ψ = sinc and the exponentials ( e − ix j · ) j ∈ Z are a Riesz basis for L ( T ), formthe inspiration for our study of this problem. More precisely, Theorems 1 and 2in [45] give conditions on families of generators ( φ α ) which allow for recovery of f ∈ P W π = V (sinc , Z ) via their interpolants in V ( φ α , Y ).Such interpolation schemes have their origin in the work of I. J. Schoenberg oncardinal interpolation via splines [59], which are related to summability methods ofthe cardinal sine series appearing in the aforementioned WKS sampling theorem.The rest of the paper is organized as follows. In the following section, we givesome preliminary definitions and discuss some of the function spaces to be studied UASI SHIFT-INVARIANT SPACES 3 in the sequel. Section 3 introduces quasi shift-invariant spaces and gives someindication of their structure as desired by Problem 1. Section 4 begins the study ofinterpolation between quasi shift-invariant spaces and gives an answer to Problem 2,part a (Theorem 2), while Section 5 provides conditions on families of interpolatinggenerators giving an answer to part b (Theorem 4 and Corollary 4). The subsequentsection provides several types of examples to illustrate the convergence phenomenondescribed previously, and it turns out that the support of the function b ψ plays animportant role in the structure of the examples. Section 7 discusses some extensionsto cardinal functions which are classical objects of study in interpolation theory,and we conclude with brief sections on inverse theorems and remarks.2. Preliminaries
Denote by L p (Ω) and ℓ p ( I ) the typical spaces of p –integrable functions over ameasurable set Ω ⊂ R and p –summable sequences indexed by the set I , respectively,with their usual norms. By L p with no set specified we mean L p ( R ), and likewise ℓ p := ℓ p ( Z ). For convenience, let ℓ ′ p := ℓ p ( Z \ { } ). Additionally, let C (Ω) be thespace of continuous functions on Ω and C ( R ) the subset of continuous functionson R vanishing at infinity.For functions f ∈ L , we will use the following normalization for the Fouriertransform: b f ( ξ ) := 1 √ π Z R f ( x ) e − iξx dx, whereby the Fourier transform can be extended to a linear isometry on L . If b f ∈ L and f is continuous, then the inversion formula is f ( x ) = √ π R R b f ( ξ ) e ixξ dξ. Plancherel’s Identity thus states that k b f k L = k f k L . We will use T to denote thetorus, which may be identified with the interval [ − π, π ). Unless otherwise specified, h· , ·i is to be taken as the inner product on L ( T ).Throughout, C will denote a constant, which may change from line to line de-pending on context, and subscripts will denote dependence upon a given parameter.Additionally, the statement k · k ≍ k · k will mean that there are constants c and c such that c k · k ≤ k · k ≤ c k · k . Function Spaces.
Throughout the sequel, we will be concerned with manyfunction spaces which arise in applications of harmonic analysis (specifically asdifferent models of the structure of signals one wishes to analyze), but which alsoenjoy use in other areas of functional analysis. The first is the classical Paley–Wiener space of bandlimited functions. Given σ >
0, let
P W σ := { f ∈ L : b f = 0 a.e. outside of [ − σ, σ ] } , endowed with the norm on L ( R ). Since P W σ is isometrically isomorphic to P W γ for any parameters σ and γ via the map J σγ : P W σ → P W γ , f ( x ) p γ/σf ( γx/σ ), in the sequel we limit our considerations to the canonical space P W π . Naturally, the Fourier transform is an isometric isomorphism from P W π to L ( T ). KEATON HAMM AND JEFF LEDFORD
Of additional utility to our analysis are various
Wiener amalgam spaces . For1 ≤ p, q ≤ ∞ , these spaces are defined by W ( L p , ℓ q ) := f : X j ∈ Z k f ( · + 2 πj ) k qL p ( T ) q < ∞ , with the suitable modification when q is infinite. We denote the norm implicit inthe definition above via k f k W ( L p ,ℓ q ) . These spaces may also be identified as the ℓ q sum of Banach spaces: ( ⊕ k ∈ Z L p ( T + 2 πk )) q , which is isometrically isomorphic to ℓ q ( L p ( T )) via the obvious map. Note that this readily implies that the amalgamspaces are Banach spaces. For the special case that is most heavily consideredin the sequel, we reduce the notation to W := W ( L ∞ , ℓ ), which is sometimescalled Wiener’s space. Note that these amalgam spaces capture both local andglobal behavior of the functions simultaneously. Loosely, functions in W ( L p , ℓ q )are locally in L p and globally in ℓ q . These spaces, first considered by Wiener, havefound great utility in harmonic analysis (see the excellent survey [34] and its manyreferences, as well as [22, 25]). As the sequel will make use of the Fourier transformsof functions in the amalgam spaces, it will be useful to adopt the convention that F V is the set of Fourier transforms of elements of V provided V ⊂ L .We will not enumerate all of the properties of these amalgam spaces, but let uscollect some facts which will be useful later on. Proposition 1. (i) If ≤ p ≤ q ≤ ∞ and ≤ r ≤ ∞ , then W ( L r , ℓ p ) ⊂ W ( L r , ℓ q ) , and k · k W ( L r ,ℓ q ) ≤ k · k W ( L r ,ℓ p ) .(ii) If ≤ r ≤ ∞ , and ≤ p ≤ q ≤ ∞ , then W ( L q , ℓ r ) ⊂ W ( L p , ℓ r ) , and k · k W ( L p ,ℓ r ) ≤ (2 π ) p − q k · k W ( L q ,ℓ r ) .(iii) W ( L p , ℓ p ) = L p .(iv) W ( L ∞ , ℓ ) ⊂ L ∩ L . (v) F W ( L ∞ , ℓ ) ⊂ C ∩ L . (vi) If f, g ∈ W ( L ∞ , ℓ ) then f g ∈ W ( L ∞ , ℓ ) .Proof. Note that (i) follows from the inclusion ℓ p ⊂ ℓ q , and the fact that the ℓ q normis subordinate to the ℓ p norm, whilst (ii) follows from the facts that L q ( T ) ⊂ L p ( T )and k · k L p ( T ) ≤ (2 π ) p − q k · k L q ( T ) . Part (iii) is evident by the definition of the norm on W ( L p , ℓ p ), and (iv) followsfrom combining (i) , (ii) and (iii) . Finally, (v) arises from (iv) and the Riemann–Lebesgue Lemma, and (vi) follows from the fact that ℓ is closed under multiplica-tion. (cid:3) For a separable Hilbert space H , let B ( H ) be the space of bounded linear opera-tors from H into itself, and recall that the strong operator topology (SOT) on B ( H )is the topology of pointwise convergence. That is, ( T n ) ⊂ B ( H ) converges in SOTto T ∈ B ( H ) provided k T n h − T h k H → h ∈ H . When the context isclear, we simply use k · k to denote the typical operator norm for elements of B ( H ).2.2. Riesz Bases and Complete Interpolating Sequences.
Two importantobjects for our subsequent study will be Riesz bases for Hilbert spaces and completeinterpolating sequences for Paley–Wiener spaces. These objects are defined in ratherdifferent ways, but are nonetheless intimately related.
UASI SHIFT-INVARIANT SPACES 5
Definition 1.
Let H be a separable, infinite dimensional Hilbert space. A family ( h j ) j ∈ Z ⊂ H is a Riesz basis for H provided it is complete and there exists a constant C ≥ (called the Riesz basis constant) such that for every finite sequence of scalars ( a j ) , (1) 1 C k a k ℓ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j a j h j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H ≤ C k a k ℓ . Equivalently, a Riesz basis is a bounded unconditional basis, or is the image ofan orthonormal basis under an invertible bounded linear operator, or is an exact,tight frame [19, 63]. Moreover, if ( h j ) is a Riesz basis for H , then each f ∈ H admits a unique representation of the form f = P j ∈ Z c j h j with ( c j ) ∈ ℓ . Definition 2.
A sequence X := ( x j ) j ∈ Z ⊂ R is a complete interpolating sequence (CIS) for P W π provided for every ( c j ) j ∈ Z ∈ ℓ , there exists a unique f ∈ P W π satisfying (2) f ( x j ) = c j , j ∈ Z . Finding a solution to (2) is called the moment problem [63]. It turns out thatthese two concepts are closely related as the following theorem shows.
Theorem 1 ([63], Theorem 9, p. 143) . A sequence X is a CIS for P W π if andonly if ( e − ix j · ) j ∈ Z is a Riesz basis for L ( T ) . Note that Definition 2 and Theorem 1 provide a natural bijection between ℓ and L ( T ) via the map ( c j ) j ∈ Z P j ∈ Z c j e − ix j · .A necessary condition for X ⊂ R to be a CIS for P W π is for it to be quasi-uniform, i.e. there are 0 ≤ q ≤ Q < ∞ such that q ≤ | x j +1 − x j | ≤ Q (assumingthat the points are ordered so that x j < x j +1 for all j ). A sufficient condition maybe found in Kadec’s 1/4–theorem [41], which states that if sup j ∈ Z | x j − j | < / X is a CIS for P W π . Consequently, CISs are available in abundance, andrestricting attention to such sequences is not overly strict. Necessary and sufficientconditions were given by Pavlov [56] in terms of zeros of certain types of entirefunctions and Muckenhoupt weights.Next, we define two operations, extension and prolongation, which will be usedextensively in the subsequent sections. First, assume that X is a CIS for P W π , andnote that on account of Theorem 1, if h ∈ L ( T ), it admits a unique representation h = P j ∈ Z c j e − ix j · in L ( T ). Therefore, (1) implies that the extension E X ( h )( t ) := X j ∈ Z c j e − ix j t , t ∈ R is locally square-integrable, and thus is well-defined almost everywhere on R . Sec-ond, for every k ∈ Z , define the prolongation operator A k X : L ( T ) → L ( T ) associ-ated with the CIS X via A k X ( h )( t ) := E X ( h )( t + 2 πk ) = X j ∈ Z c j e − ix j ( t +2 πk ) , t ∈ T . Note A k X is not merely translation, as it is viewed as an operator mapping into L ( T ), so is better viewed as an operation on the coefficients. An application of(1) implies that k A k X k ≤ C X , k ∈ Z , KEATON HAMM AND JEFF LEDFORD where C X is the Riesz basis constant for the exponential system associated with X as in Definition 1. The same bound holds for the adjoint, A ∗ k X . Consequently, thefamilies ( A k X ) , ( A ∗ k X ) are uniformly bounded subsets of B ( L ( T )).In the sequel, we will distinguish between prolongation operators for multipleCISs, and if Y is another CIS, then E Y and A k Y are defined analogously via theexpansion of h in terms of the Riesz basis ( e − iy j · ) j ∈ Z .Finally, each Riesz basis has with it an associated dual Riesz basis. Indeed, if( e e j ) j ∈ Z are the coordinate functionals associated with the basis ( e − ix j · ) j ∈ Z , i.e. thedual elements such that (cid:10) e e j , e − ix k · (cid:11) = δ j,k , then any h ∈ L ( T ) admits a uniquerepresentation in each basis as follows: h = X j ∈ Z (cid:10) h, e − ix j · (cid:11) e e j = X j ∈ Z h h, e e j i e − ix j · . These coordinate functionals are the dual Riesz basis. It should be noted that thefunctions e e j need not be continuous in general, though if x j = j for every j , thenevidently e e j = e − ij · .Our use of the prolongation and extension operators above stem from Lyubarskiiand Madych [48], who additionally provide a pleasant extension of the classicalPoisson Summation Formula to sequences which are CISs for P W π [49].3. Quasi Shift-Invariant Spaces and their Structure
Let us now introduce the primary object of the subsequent study: the so-called quasi shift-invariant spaces (so-named in [26]). For the moment, suppose that X := ( x j ) j ∈ Z ⊂ R is a discrete sequence, i.e. inf j = k | x j − x k | >
0, and that ψ ∈ L .Then define V ( ψ, X ) := span { ψ ( · − x j ) : j ∈ Z } , where the closure is taken in L ( R ). In the special case X = Z , which is one of theprimary motivations of this study, the space V ( ψ ) := V ( ψ, Z ) is called the principalshift-invariant space associated with the generator ψ (which is also commonly calledthe kernel or window function; our use of the term generator follows from [4]). Foran excellent survey of the weighted sampling problem in shift-invariant spaces, thereader is invited to consult [4]. Let us note that in the case that { ψ ( · − x j ) : j ∈ Z } is a Riesz basis for V ( ψ, X ), then the space can be described in a manner that iseasier to handle in practice. Namely,(3) V ( ψ, X ) = X j ∈ Z c j ψ ( · − x j ) : ( c j ) ∈ ℓ , where convergence of the series is taken to be in L . Indeed, in much of theliterature, the space is defined by (3) (for example, [4, 26]).In light of the fact that quasi shift-invariant spaces are generalizations of shift-invariant spaces, it is natural to ask how similar their structure is. Consequently,there are some natural questions that present themselves here, the first of which is:when is V ( ψ, X ) a closed subspace of L ? Secondly, when is { ψ ( · − x j ) : j ∈ Z } aRiesz basis for V ( ψ, X )? The third question of interest to us (Problem 2) is that ofwhen the nonuniform sampling problem is well-posed, i.e. when is f uniquely andstably determined by its values ( f ( y j )) for some discrete Y ⊂ R ? This of course UASI SHIFT-INVARIANT SPACES 7 requires pointwise evaluation to be well-defined in the space V ( ψ, X ). The answers,as will be demonstrated, are similar to the shift-invariant space case.3.1. Regular Generators.
We now turn to some regularity conditions on thegenerator ψ and the translation set X , and elucidate the structure of the associatedquasi shift-invariant spaces. Henceforth we assume that ψ enjoys the followingproperties:(A1) ψ ∈ L ( R ) is positive definite, and(A2) b ψ ∈ W ( L ∞ , ℓ ) such that C ψ := P j =0 k b ψ ( · + 2 πj ) k L ∞ ( T ) inf ξ ∈ T | b ψ ( ξ ) | < ∞ For convenience later on, if ψ satisfies (A1) and (A2), then we define(4) δ ψ := inf ξ ∈ T | b ψ ( ξ ) | , which is necessarily positive.Note that by Proposition 1, (A2) implies that the generator ψ is continuous.Also, in the case of V ( ψ, Z ), properties (A1) and (A2) imply that { ψ ( · − j ) : j ∈ Z } is a Riesz basis for its closed linear span, and consequently that V ( ψ, Z ) is a closedsubspace of L . Additionally, these assumptions are not overly strict in that theyencompass the well-studied sinc kernel, which forms the launching point of ourstudy (and many others, for that matter). Moreover, the literature on interpola-tion and approximation of functions by positive definite kernels is extensive, andemploys connections with reproducing kernel Hilbert spaces, polynomial splines,and other techniques. See, for example, [53], and for many references on scattered-data approximation, [64].For the duration of this work, we will make the assumption that the shift sequence X is a CIS for P W π , or equivalently, via Theorem 1, that it contains the frequenciesof a Riesz basis of exponentials for L ( T ). This restriction, while a special case of thegeneral quasi shift-invariant space, comes from the motivation of the interpolationproblem found in Theorem 1 of [45]. Throughout this section, we will use theequivalent definition in (3), which is justified on account of the following proposition,which is the main structural statement on quasi shift-invariant spaces in this section. Proposition 2. If X is a CIS for P W π and ψ satisfies (A1) and (A2) , then thefollowing hold:(i) { ψ ( · − x j ) : j ∈ Z } is a Riesz basis for V ( ψ, X ) , and moreover, δ ψ C X k c k ℓ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ∈ Z c j ψ ( · − x j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) ≤ C X k b ψ k W ( L ∞ ,ℓ ) k c k ℓ ; (ii) V ( ψ, X ) is a closed subspace of L ;(iii) V ( ψ, X ) ⊂ C ∩ L ( R ) .(iv) For all f ∈ V ( ψ, X ) , k b f k L ( T ) ≤ k f k L ( R ) ≤ (cid:0) C X C ψ (cid:1) k b f k L ( T ) . Proof.
Note that (ii) follows immediately from (i) . Item (i) is [35, Theorem 2.4];since we use similar periodization arguments often in the sequel, we choose to givethe proof here. Since there is only one CIS of interest, let A k = A k X for ease of KEATON HAMM AND JEFF LEDFORD notation in the remainder of the proof. In light of Plancherel’s Identity, it sufficesto check the Riesz basis inequality in (i) with the middle term replaced by the normof its Fourier transform. To wit, notice that Z R | b ψ ( ξ ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ Z c j e − ix j ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ = X k ∈ Z Z T | b ψ ( ξ + 2 πk ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A k X j ∈ Z c j e − ix j ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≤ k b ψ k W ( L ∞ ,ℓ ) C X k c k ℓ , where we have used periodization, the operator bound on A k , the Riesz basis in-equality, and the fact that k ·k ℓ ≤ k ·k ℓ . The above inequality gives the right-handside of the stated Riesz basis inequality for { ψ ( · − x j ) : j ∈ Z } . To prove the lowerbound, simply note that Z R | b ψ ( ξ ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ Z c j e − ix j ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≥ Z T | b ψ ( ξ ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ Z c j e − ix j ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ ≥ δ ψ C X k c k ℓ , which is the desired lower bound.In light of the Riemann–Lebesgue Lemma, statement (iii) only requires thatwe check that the Fourier transform of f ∈ V ( ψ, X ) is in L ∩ L ( R ). That b f issquare-integrable follows from (i) , whereas the calculation to show that b f ∈ L follows by the same method, and so is omitted. It should be noted that while theabove calculation only requires b ψ ∈ W ( L ∞ , ℓ ), showing that b f ∈ L requires that b ψ ∈ W ( L ∞ , ℓ ), which is (A2).The first inequality in (iv) is a trivial consequence of Plancherel’s Identity. Thesecond inequality depends upon Plancherel’s Identity and periodization. Let f = P c j ψ ( · − x j ), then Z R | f ( x ) | dx = Z R | b f ( ξ ) | dξ = Z T | b f ( ξ ) | dξ + X k =0 Z T | b f ( ξ + 2 πk ) | dξ = Z T | b f ( ξ ) | dξ + X k =0 Z T (cid:12)(cid:12)(cid:12) b ψ ( ξ + 2 πk ) A k (cid:16)X c j e ix j · (cid:17) ( ξ ) (cid:12)(cid:12)(cid:12) dξ ≤ k b f k L ( T ) + C X C ψ k b f k L ( T ) . The inequality follows from elementary estimates, the uniform bound on the prolon-gation operators A k , multiplying and dividing by b ψ , and the fact that k·k ℓ ≤ k·k ℓ . (cid:3) It should be noted that Proposition 2 and its proof imply that F V ( ψ, X ) ⊂ ∩ L ,and that the Fourier inversion formula holds for functions in V ( ψ, X ). Note also,from (iv) above, that V ( ψ, X ) is isomorphic to P W π if X is a CIS for P W π . UASI SHIFT-INVARIANT SPACES 9
Remark 1.
It should be noted that, under more relaxed assumptions on ψ , itstranslates need not form a Riesz basis for V ( ψ, X ) . Indeed, if b ψ ∈ C ( T ) , but supp( b ψ ) ( T , then { ψ ( · − x j ) : j ∈ Z } might fail to be a Riesz basis, or even aframe for its closed linear span (see [19] for the definition of a frame). Interestingly, the support of b ψ plays an important role in the structure of thesespaces under the assumptions above as the following proposition shows. Proposition 3.
Suppose ψ satisfies (A1) and (A2) . Then(i) If X is a CIS for P W π and supp( b ψ ) = T , then V ( ψ, X ) = P W π .(ii) If supp( b ψ ) ) T , then there exist CISs for P W π , X and Y , with X 6 = Y and V ( ψ, X ) = V ( ψ, Y ) .Proof. Proof of (i) : Since the Fourier transform is an isometric isomorphism be-tween
P W π and L ( T ), it suffices to show that F V ( ψ, X ) = L ( T ). Since ( e − ix j · ) j ∈ Z is a Riesz basis for L ( T ), we have that F V ( ψ, X ) = { b ψQ : Q ∈ L ( T ) } , which is L ( T ) since b ψ ( ξ ) ≥ δ ψ > T .Proof of (ii) : Suppose that Y = Z and X = Z \ { } ∪ {√ } , and let f = ψ ( · − √ ∈ V ( ψ, X ). Then b f = b ψe − i √ · . However, any g ∈ V ( ψ, Z ) satisfies b g = b ψQ where Q is a 2 π –periodic function. Consequently, b f and b g must differ in L ( T + 2 πk ) for all but a single k ∈ Z . (cid:3) The simple example of item (ii) turns out to be an important one and will beused again to discuss the limitations of the interpolation and recovery schemesdeveloped in subsequent sections. Further discussion of the case when supp( b ψ ) ) T is postponed until Section 6.A natural question is how large can the subspaces V ( ψ, X ) be? Can we obtainall of L ( R ) in this manner? Such questions have been considered for general p andtranslation sets X [6, 23, 55, 54]. The answer is essentially that translations canspan L and L p for 2 < p < ∞ ; however, one cannot generate an unconditionalbasis for all of L p ( R ) in this manner for any 1 ≤ p ≤ ∞ .4. Interpolation
Having elucidated some of the structure of the quasi shift-invariant spaces above,we now turn to the task of interpolation of functions in one such space from another,with the ultimate goal of classifying how well interpolation of functions in a targetspace V ( ψ, X ) with functions in V ( φ, Y ) performs.Henceforth, we will fix a particular choice of ψ satisfying (A1) and (A2) and aCIS X , and we wish to interpolate data sampled from V ( ψ, X ) at some sequence Y . We require that our interpolant have the form(5) I Y φ f ( x ) = X j ∈ Z a j φ ( x − y j ) , (that is, I Y φ f ∈ V ( φ, Y )) where Y := ( y j ) is a CIS for P W π and ( a j ) are theinterpolating coefficients. Here, φ is the interpolation generator .On the way, we first demonstrate that the sampling problem is indeed well-posedas long as one samples at a CIS: Lemma 1.
Let X and Y be CISs for P W π , and suppose ψ satisfies (A1) and (A2) .If f ∈ V ( ψ, X ) with f = P n ∈ Z c n ψ ( · − x n ) , then ( f ( y j )) ∈ ℓ . Moreover, k ( f ( y j )) k ℓ ≤ √ π C X C Y (cid:16) δ − ψ k b ψ k L ∞ ( T ) + C ψ (cid:17) k f k L ( R ) . Proof.
We begin by noting that pointwise evaluation is well-defined by Proposition2 (iii) . Recalling that h· , ·i is the inner product on L ( T ), we have f ( y j ) = 1 √ π Z R b f ( ξ ) e iy j ξ dξ = 1 √ π Z R b ψ ( ξ ) X n ∈ Z c n e − ix n ξ ! e iy j ξ dξ = 1 √ π X k ∈ Z Z T b ψ ( ξ + 2 πk ) A k X X n ∈ Z c n e − ix n · ! ( ξ ) A k Y ( e − iy j · )( ξ ) dξ = 1 √ π *X k ∈ Z A ∗ k Y " b ψ ( · + 2 πk ) A k X X n ∈ Z c n e − ix n · ! , e − iy j · + . Now we may use the Riesz basis inequality for ( e − iy j · ), which yields k ( f ( y j )) k ℓ ≤ √ π C Y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z A ∗ k Y " b ψ ( · + 2 πk ) A k X X n ∈ Z c n e − ix n · ! L ( T ) . Using the triangle inequality and the bound for A ∗ k Y and A k X provides the upperbound 1 √ π C X C Y k b ψ k W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X n ∈ Z c n e − ix n · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( T ) , which we may bound above in terms of k f k L ( R ) . Multiplying and dividing by b ψ provides us with the bound1 √ π C X C Y δ − ψ k b ψ k W k b f k L ( T ) = 1 √ π C X C Y (cid:16) δ − ψ k b ψ k L ∞ ( T ) + C ψ (cid:17) k b f k L ( T ) ≤ √ π C X C Y (cid:16) δ − ψ k b ψ k L ∞ ( T ) + C ψ (cid:17) k f k L ( R ) , which completes the proof. (cid:3) A similar argument allows us to find the following bound in terms of k c k ℓ : k ( f ( y j )) k ℓ ≤ √ π C X C Y k b ψ k W k c k ℓ . Existence of Interpolants.
The following theorem demonstrates that inter-polation is well-defined for a large variety of generators.
Theorem 2.
Suppose that X and Y are CISs for P W π , ψ satisfies (A1) and (A2),and f ∈ V ( ψ, X ) . If φ satisfies (A1) and (A2) , then there exists a unique sequence ( a j ) ∈ ℓ such that the function I Y φ f of (5) satisfies:(i) I Y φ f ( y k ) = f ( y k ) for all k ∈ Z , and(ii) I Y φ f ∈ C ∩ L ( R ) . UASI SHIFT-INVARIANT SPACES 11
Proof.
This is essentially a reformulation of Corollary 1 and Proposition 1 in [45]since (A1) and (A2) imply the conditions found there. (cid:3)
We note the following bound on the interpolating coefficients as in Theorem 2:(6) k ( a j ) k ℓ ≤ C X (cid:16) k b φ k L ∞ ( T ) δ − φ + C X C φ (cid:17) k ( f ( y j )) k ℓ . Note that Theorem 2 implies that the interpolation operator I Y φ is a bounded lin-ear operator from V ( ψ, X ) → C ∩ L ( R ). See also [29, 47] for similar interpolationresults in higher dimensions.This interpolation theorem provides us with the following norm equivalences inthe quasi shift-invariant space. Theorem 3.
Suppose ψ satisfies (A1) and (A2) , and that X is a CIS for P W π .Then if f = P j ∈ Z c j ψ ( · − x j ) ∈ V ( ψ, X ) , we have k f k L ≍ k ( c j ) k ℓ ≍ k ( f ( x j )) k ℓ . Proof.
The equivalence k f k L ≍ k ( c j ) k ℓ follows from Proposition 2(i). Meanwhile,the fact that k ( c j ) k ℓ ≤ C k ( f ( x j )) k ℓ follows from (6) when φ = ψ and Y = X . Inparticular, this uses the fact that b ψ ∈ W ( L ∞ , ℓ ) implies that the bi-infinite matrix( ψ ( x j − x k )) j,k ∈ Z defines an operator in B ( ℓ ) (which in turn relies on the fact that X is a CIS). Finally, that k ( f ( x j )) k ℓ ≤ C k ( c j ) k ℓ follows from the proof of Lemma1. Putting together these estimates yields the conclusion of the theorem. (cid:3) Note that the following stems directly from the above theorem.
Corollary 1. If ψ and X are as in Theorem 3, then for any φ satisfying (A1) and (A2) , the interpolation operator I X φ : V ( ψ, X ) → V ( φ, X ) is boundedly invertible.Moreover, the sampling operator S X : V ( ψ, X ) → ℓ associated with X defined by f ( f ( x j )) j ∈ Z is an isomorphism. Consequently, V ( ψ, X ) and V ( ψ, Y ) are isomorphic to each other if both X and Y are CISs for P W π . However, they are, in general, not the same subspace of L as Proposition 3 (ii) shows. It should also be noted that in the case φ = ψ but X 6 = Y , interpolation is delicate. In particular, Baxter and Sivakumar [9] showedthat if ψ ( x ) = e −| x | , X = Z , and Y = Z + , then the interpolation operator I Y ψ : V ( ψ, X ) → V ( ψ, Y ) is not boundedly invertible.Gr¨ochenig and St¨ockler [26] and recently together with Romero [27] give moregeneral sampling results similar to the second part of Corollary 1 when ψ is a totallypositive function, but for quasi-uniform X . Moreover, their techniques, using Gaborframe analysis, are quite different than the ones used here, and have some ratherinteresting implications.4.2. Bounds for the Interpolation Operators.
In this subsection, we exploresome properties of the interpolation operators I Y φ : V ( ψ, X ) → V ( φ, Y ) with thetwo-fold aim of extracting information about general interpolation properties be-tween quasi shift-invariant spaces and setting the stage for the recovery results inthe sequel.In what follows, let X and Y be fixed, but arbitrary CISs for P W π , and recallthat, given a function φ satisfying (A1) and (A2), each f ∈ V ( ψ, X ) has a uniqueinterpolant I Y φ f ∈ V ( φ, Y ) via Theorem 2 which satisfies I Y φ f ( y k ) = f ( y k ), k ∈ Z . To move forward, we define some auxiliary operators whose importance will berevealed shortly. We begin with a simple multiplication operator: for g ∈ L ( T ),let M φ g := δ φ b φ g, where δ φ is defined as in (4). Given a function φ and an integer k , we denotemultiplication by δ − φ b φ ( · + 2 πk ) by T φ,k ; that is, for g ∈ L ( T ), T φ,k g := δ − φ b φ ( · + 2 πk ) g. First, let us note that the conditions (A1) and (A2) guarantee that these are welldefined, bounded operators on L ( T ). Indeed we have the transparent bounds k M φ k ≤ , and k T φ,k k ≤ C φ if k = 0, with k T φ, k ≤ δ − φ k b φ k L ∞ ( T ) , whence k T φ,k k is boundedindependent of k . Of particular note is that (A2) implies that(7) X k =0 k T φ,k g k L ( T ) ≤ C φ k g k L ( T ) . Finally, let B φ , e B φ : L ( T ) → L ( T ) be defined by B φ g := X k =0 A ∗ k Y (cid:2) T φ,k A k X g (cid:3) and e B φ g := X k =0 A ∗ k Y (cid:2) T φ,k A k Y g (cid:3) . Via the same method of calculation found in the previous proofs, one obtainsthe following bounds for these operators:(8) k B φ k ≤ C Y C X C φ , and(9) k e B φ k ≤ C Y C φ . Note that e B φ , T φ,k , and M φ are positive (i.e. h M φ g, g i ≥ g ∈ L ( T ));however, B φ is not positive in general.We simplify the notation of the interpolants by naming the nonharmonic Fourierseries that arises in their Fourier transform: d I Y φ f ( ξ ) = X j ∈ Z a j e − iy j ξ b φ ( ξ ) =: u ( ξ ) b φ ( ξ ) . The following lemma will be exploited frequently.
Lemma 2.
Under the assumptions of Theorem 2, let f ∈ V ( ψ, X ) , and I Y φ f be itsunique interpolant in V ( φ, Y ) . The following equality holds almost everywhere on T : ( I + B ψ M ψ ) b f = ( I + e B φ M φ ) d I Y φ f , where I denotes the identity operator on L ( T ) . UASI SHIFT-INVARIANT SPACES 13
Proof.
The proof uses periodization and the fact that Y is a CIS. Knowing that f ( y j ) = I Y φ f ( y j ), we simply expand these in terms of the Fourier transform. First,note that √ πI Y φ f ( y j ) = Z R b φ ( ξ ) u ( ξ ) e iy j ξ dξ = X k ∈ Z Z T b φ ( ξ + 2 πk ) A k Y u ( ξ ) A k Y ( e − iy j · )( ξ ) dξ = X k ∈ Z Z T A ∗ k Y hb φ ( · + 2 πk ) A k Y u i ( ξ ) e − iy j ξ dξ = *X k ∈ Z A ∗ k Y h b φ ( · + 2 πk ) A k Y u i , e − iy j · + = D ( I + e B φ M φ ) d I φ f , e − iy j · E , where the inner product is on L ( T ).Similarly, we have √ πf ( y j ) = *X k ∈ Z A ∗ k Y " b ψ ( · + 2 πk ) A k X X n ∈ Z c n e ix n · ! , e − iy j · + = * b f + X k =0 A ∗ k Y " b ψ ( · + 2 πk ) A k X X n ∈ Z c n e ix n · ! , e − iy j · + = D ( I + B ψ M ψ ) b f , e − iy j · E . Thus the conclusion of Lemma 1, the fact that Y is a CIS, and that the equalitiesabove hold for all j ∈ Z completes the proof. (cid:3) The following results illustrate the nature of the interpolation operators betweentwo quasi shift-invariant spaces.
Proposition 4.
Under the assumptions of Theorem 2, the following holds: k d I Y φ f k L ( T ) ≤ (1 + C Y C φ ) k ( I + B ψ M ψ ) b f k L ( T ) , f ∈ V ( ψ, X ) . Consequently, k d I Y φ f k L ( T ) ≤ (1 + C Y C φ )(1 + C X C Y C ψ ) k b f k L ( T ) , f ∈ V ( ψ, X ) . Proof.
Note that Lemma 2, the triangle inequality, and the fact that M φ d I Y φ f = δ φ u provide us with the estimate k d I Y φ f k L ( T ) ≤ k ( I + B ψ M ψ ) b f k L ( T ) + δ φ k e B φ u k L ( T ) . Now the second term is majorized by δ φ C Y C φ k u k L ( T ) on account of (9), whenceit suffices to show that(10) δ φ k u k L ( T ) ≤ k ( I + B ψ M ψ ) b f k L ( T ) . Rewriting the result of Lemma 2 as b φu + δ φ e B φ u = ( I + B ψ M ψ ) b f , then taking the inner product with u and appealing to (A1) and (A2), the Cauchy-Schwarz inequality, and positivity of e B φ , yields h b φu, u i ≤ k ( I + B ψ M ψ ) b f k L ( T ) k u k L ( T ) . Now the elementary inequality δ φ k u k L ( T ) ≤ h b φu, u i combined with the previousestimate yields (10), which completes the proof of the first inequality. The secondstatement follows directly from the fact that k M ψ k ≤ (cid:3) Using these bounds, we may now estimate the operator norm of the interpolationoperator I Y φ : V ( ψ, X ) → V ( φ, Y ). Corollary 2.
With the notation and assumptions above, the following holds for all f ∈ V ( ψ, X ) : k I Y φ f k L ( R ) ≤ (cid:0) C Y C φ (cid:1) (1 + C Y C φ ) (1 + C X C Y C ψ ) k b f k L ( T ) . Proof.
Since I Y φ f ∈ V ( φ, Y ), we simply combine Propositions 2 (iv) and 4. (cid:3) Recovery Criteria
Having determined when interpolation of functions in V ( ψ, X ) is possible viafunctions in V ( φ, Y ) in Section 4, we now turn to some approximate samplingschemes which allow for recovery of f ∈ V ( ψ, X ) in a limiting sense from its inter-polants in a family of spaces ( V ( φ α , Y )) α ∈ A . The idea is that while the generator ψ may be complicated, or decay slowly as sinc does, it may be replaced by aninterpolating generator φ α which gives approximate recovery in both L and L ∞ ,but which may have a much simpler structure. More specifically, we consider thefollowing problem: Problem 3.
Given ψ satisfying (A1) and (A2) and a CIS X , find conditions ona family of interpolating generators Φ := ( φ α ) α ∈ A and CISs Y such that for every f ∈ V ( ψ, X ), its interpolants I Y φ α f ∈ V ( φ α , Y ) converge to f in L and uniformly.5.1. Preliminaries.
With the above considerations in mind, consider the followingcriteria:(B1) For every α ∈ A , φ α satisfies (A1) and (A2).(B2) C Φ := sup α ∈ A C φ α < ∞ , where C φ α is as in condition (A2).(B3) lim α →∞ P k =0 k T ψ,k A k X ( M ψ − M φ α ) g k L ( T ) = 0 for every g ∈ L ( T ).(B4) lim α →∞ P k =0 k ( T φ α ,k A k Y − T ψ,k A k X ) M φ α g k L ( T ) = 0 for every g ∈ L ( T ).Let us stress that while (B3) and (B4) may seem rather abstruse at the moment,stronger hypotheses may be used which imply these conditions but which nonethe-less give rise to many examples and additionally are more easily verified in practice.We will discuss these in more detail in Section 6; presently we turn our attentionto consequences of these criteria. Note that if X = Y = Z , then these conditionsare much easier to handle since A k X and A k Y are the identity on L ( T ).Condition (B1) implies that for all α , M φ α ∈ B ( L ( T )), and additionally(11) k M φ α k ≤ , α ∈ A. Together, (B1) and (B2) show that if k = 0, T φ α ,k ∈ B ( L ( T )), with k T φ α ,k k ≤ C φ α ≤ C Φ , and thus are bounded independently of α and k . Moreover, (7) implies UASI SHIFT-INVARIANT SPACES 15 that X k =0 k T φ α ,k g k L ( T ) ≤ C Φ k g k L ( T ) , α ∈ A. Likewise, we have k B φ α k ≤ C Y C X C Φ and k e B φ α k ≤ C Y C Φ .If k = 0, (A2), and hence (B1), provides the bound k T φ α , k ≤ δ − φ α k c φ α k L ∞ ( T ) ;however this bound could well depend on α , as will be made more clear in theexamples section that follows.5.2. Recovery.
With these notions in hand, we are ready to demonstrate ourmain recovery results. Let us first note that Lemma 2 and Proposition 4 imply thefollowing.
Corollary 3.
The operators ( I + e B φ α M φ α ) : L ( T ) → L ( T ) are invertible and thenorms of the inverses are bounded independent of α . Additionally, we have the following.
Lemma 3. (B1)–(B4) imply that e B φ α M φ α − B ψ M ψ → in the SOT on B ( L ( T )) .Proof. Let g ∈ L ( T ). Then k ( e B φ α M φ α − B ψ M ψ ) g k L ( T ) ≤ C Y X k =0 k ( T φ α ,k A k Y M φ α − T ψ,k A k X M ψ ) g k L ( T ) ≤ C Y X k =0 k ( T φ α ,k A k Y − T ψ,k A k X ) M φ α g k L ( T ) + C Y X k =0 k T ψ,k A k X ( M φ α − M ψ ) g k L ( T ) , and both terms converge to 0 as α → ∞ on account of (B3) and (B4). The firstinequality above stems from the uniform bounds on the prolongation operators A k Y and their adjoints. (cid:3) It should be noted that in general, M φ α need not converge to M ψ in the SOTon B ( L ( T )). An example of this is provided by regular interpolators discussed inSection 6. We may now prove our main recovery result. Theorem 4.
Suppose that X and Y are CISs for P W π , ψ satisfies (A1) and (A2) ,and ( φ α ) α ∈ A satisfies (B1)–(B4) . Then for every f ∈ V ( ψ, X ) , lim α →∞ k f − I Y φ α f k L ( R ) = 0 , where I Y φ α f is the unique element of V ( φ α , Y ) which interpolates f at Y .Proof. Plancherel’s Identity allows us to check this result in the Fourier domain;thus we estimate k b f − [ I Y φ α f k L ( R ) ≤k b f − [ I Y φ α f k L ( T ) + k b f − [ I Y φ α f k L ( R \ T ) = : I + I . We begin with I . On T , we have b f − [ I Y φ α f = (cid:16) I − ( I + e B φ α M φ α ) − ( I + B ψ M ψ ) (cid:17) b f =( I + e B φ α M φ α ) − ( e B φ α M φ α − B ψ M ψ ) b f . Therefore, Corollary 3 and Lemma 3 imply that I → α → ∞ . Next, notice that I is majorized by X k =0 k T ψ,k A k X M ψ b f − T φ α ,k A k Y M φ α [ I Y φ α f k L ( T ) , which, in turn, is majorized by I , + I , + I , , where I , := X k =0 k T ψ,k A k X M ψ ( b f − [ I Y φ α f ) k L ( T ) ,I , := X k =0 k T ψ,k A k X ( M ψ − M φ α ) [ I Y φ α f k L ( T ) , and I , := X k =0 k ( T ψ,k A k X − T φ α ,k A k Y ) M φ α [ I Y φ α f k L ( T ) . Notice that I , ≤ C X C ψ k b f − [ I Y φ α f k L ( T ) , which converges to 0 as this is the I term above. Subsequently, I , is majorized by X k =0 k T ψ,k A k X ( M ψ − M φ α )( [ I Y φ α f − b f ) k L ( T ) + X k =0 k T ψ,k A k X ( M ψ − M φ α ) b f k L ( T ) , where the first term is majorized by a constant multiple of k b f − [ I Y φ α f k L ( T ) , whichis I , hence converges to 0, and the second term converges to 0 as a result of (B3).Finally, I , is bounded by X k =0 k ( T ψ,k A k X − T φ α ,k A k Y ) M φ α ( [ I Y φ α f − b f ) k L ( T ) + X k =0 k ( T ψ,k A k X − T φ α ,k A k Y ) M φ α b f k L ( T ) . The first term is bounded above by a constant multiple of k b f − [ I Y φ α f k L ( T ) bya similar argument to the first term related to I , , hence converges to 0. Thesecond term converges to 0 on account of (B4). Putting these estimates together,we conclude that I → α → ∞ , whence the conclusion of the theorem. (cid:3) Corollary 4.
With the notations and assumptions of Theorem 4, lim α →∞ | f ( x ) − I Y φ α f ( x ) | = 0 uniformly on R for every f ∈ V ( ψ, X ) .Proof. The proof follows directly from the proof of Theorem 4 by noticing that k f − I Y φ α f k L ∞ ( R ) ≤ k b f − [ I Y φ α f k L ( R ) , which by periodization and the Cauchy-Schwarzinequality, is majorized by a constant multiple of k b f − [ I Y φ α f k L ( T ) + X k =0 k T ψ,k A k X M ψ b f − T φ α ,k A k Y M φ α [ I Y φ α f k L ( T ) . The first term above converges to 0 by Theorem 4, whilst the second is handled asin the proof thereof. (cid:3)
UASI SHIFT-INVARIANT SPACES 17
For additional convergence phenomena similar to the ones listed in this section,the interested reader is referred to [28, 29, 32, 45, 47, 48, 51, 61].6.
Examples
Here we will provide a few examples of the phenomena described above. Suchconcrete considerations will also lead us to some alternative statements of the cri-teria (B3) and (B4) which may be more readily checked in practice. Our exam-ples are broken up into three cases based on the support of b ψ . The first is whensupp( b ψ ) = T (note that condition (A2) implies that supp( b ψ ) ⊇ T ). The secondcase is supp( b ψ ) ⊂ [ − A, A ] for some π < A < ∞ , and the final case is when thesupport is not contained in any bounded interval.6.1. Case supp( b ψ ) = T – Regular Interpolators. To begin, let us demonstratethat the more general setting here indeed recovers the specific considerations of[45]; namely, for the case of V (sinc , X ) = P W π , the criteria (B1)–(B4) imply thecriteria therein of so-called regular interpolators .A family ( φ α ) α ∈ A is said to be a family of regular interpolators for P W π provided(B1) and (B2) are satisfied, and in addition,(12) lim α →∞ δ φ α c φ α = 0 , a.e. on T . Note that, in particular, (12) implies that M φ α → B ( L ( T )) as α → ∞ on account of (11) and the dominated convergence theorem. Theorem 5. If ( φ α ) is a family of regular interpolators for P W π and ψ satisfies (A1) and (A2) and supp( b ψ ) = T , then ( φ α ) α ∈ A satisfies (B1)–(B4) . In particular,Theorem 4 and Corollary 4 are valid for such families.Proof. By definition, (B1) and (B2) are satisfied. Notice also that condition (B3)is vacuous because of the assumption on the support of b ψ . Finally, by the nowfamiliar argument, (B4) may be shown as follows: X k =0 k T φ α ,k A k Y M φ α g k L ( T ) ≤ C Φ C Y k M φ α g k L ( T ) , which tends to 0 as α → ∞ via the observation that M φ α → B ( L ( T )). (cid:3) Consequently, Theorem 5 recovers Theorem 1 of [45], which constitutes the spe-cial case when ψ is the sinc function, whose Fourier transform is the characteristicfunction of T . Also regard that the proof follows from Proposition 3 (i) in this casegiven the assumptions on ψ . Examples of regular interpolators may be found inSection 5 of [45] and Section 8 of [28], but here we note that a prominent exampleis the family of Gaussian generators: φ α ( x ) = e −| x/α | , α ≥ A Class of Convolution Examples for Bandlimited ψ . Here, we focus onthe special case when ψ is a generator satisfying (A1) and (A2) such that supp( b ψ ) ⊂ [ − A, A ] for some π < A < ∞ . Note that if b ψ ∈ W ( L ∞ , ℓ ) and b ψ is compactlysupported on [ − A, A ], then ψ ∈ P W A . However, not everything in P W A is theFourier transform of a function in W ( L ∞ , ℓ ) . Indeed, consider a function f suchthat b f is unbounded on T , e.g. f where b f ( ξ ) = ξ − / χ (0 ,π ) ( ξ ) . This f is in P W π because its Fourier transform is square-integrable, but b f fails to be in the amalgamspace.Additionally, we assume that the interpolation set Y coincides with the set X defining the space V ( ψ, X ) (the reason for this is explained in Section 6.4). Supposethat ( φ α ) α ∈ A is a family of generators, each of which satisies (A1) and (A2). Let τ α := φ α ∗ ψ, and note that Proposition 1 implies that the convolution theorem holds, i.e. c τ α = c φ α b ψ . Then we have the following: Proposition 5.
Suppose ψ satisfies (A1) and (A2) , supp( b ψ ) ⊂ [ − A, A ] for some π < A < ∞ , and Y = X is a CIS for P W π . Let N := ⌈ π A ⌉ . If ( φ α ) α ∈ A satisfies (B1) , (B2’) sup α ∈ A δ − φ α k c φ α k W ( L ∞ ,ℓ ( {− N,..., − , ,...,N } ) ≤ C, and (B3’) lim α →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) δ − φ α c φ α − δ − φ α δ − ψ δ − τ α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W ( L ∞ ,ℓ ( {− N,...,N } )) = 0 , then ( τ α ) α ∈ A satisfies (B1)–(B4) , where τ α = φ α ∗ ψ . Consequently, Theorem 4and Corollary 4 hold for ( τ α ) α ∈ A .Proof. Note that (A1) for τ α follows from the fact that (A1) holds for φ α and ψ .For (A2), note that(13) δ τ α ≥ δ φ α δ ψ . Thus, to check (B2) for ( τ α ), we need only notice that k c τ α ( · + 2 πk ) k L ∞ ( T ) ≤k c φ α ( · + 2 πk ) k L ∞ ( T ) k b ψ ( · + 2 πk ) k L ∞ ( T ) , whence applying (13) and the fact that for ℓ sequences, k ab k ℓ ≤ k a k ℓ k b k ℓ , yields δ − τ α k c τ α k W ′ ≤ δ − φ α δ − ψ k c φ α k W ′ k b ψ k W ′ , which is bounded by a constant C independent of α on account of the fact that( φ α ) satisfies (B2’) (here we have abbreviated the amalgam space in question to W ′ for brevity).Next we check (B3) and (B4). Notice that | δ − τ α c τ α − δ − ψ b ψ | = | δ − τ α c φ α b ψ − δ − ψ b ψ | = | b ψ | δ − τ α (cid:12)(cid:12)(cid:12)(cid:12)c φ α − δ − ψ δ − τα (cid:12)(cid:12)(cid:12)(cid:12) ≤ | b ψ | δ − φ α δ − ψ (cid:12)(cid:12)(cid:12)(cid:12)c φ α − δ − ψ δ − τα (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ − ψ | b ψ | (cid:12)(cid:12)(cid:12)(cid:12) δ − φ α c φ α − δ − φα δ − ψ δ − τα (cid:12)(cid:12)(cid:12)(cid:12) , where the first inequality follows from (13).Consequently, letting W := W ( L ∞ , ℓ ( {− N, . . . , N } )), k δ − τ α c τ α − δ − ψ b ψ k W ≤ δ − ψ k b ψ k W (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) δ − φ α c φ α − δ − φ α δ − ψ δ − τ α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W . UASI SHIFT-INVARIANT SPACES 19
To conclude the proof, it suffices to notice that the above inequality together with(B3’) implies both (B3) and (B4) for ( τ α ). In particular, (B3’) implies that M τ α → M ψ in the SOT on B ( L ( T )). (cid:3) Remark 2.
As a special case of this, suppose that δ τ α = δ φ α δ ψ , which can happene.g. if c φ α and b ψ are even and decreasing on [0 , π ] . In this case, (B3’) reduces tothe statement that k δ − φ α c φ α − k W ( L ∞ ,ℓ ( {− N,...,N } )) → as α → ∞ . Remark 3.
It should also be noticed that a family ( φ α ) satisfying (B2’) need notsatisfy (B2) as the following example will illustrate. That is, δ − φ α k c φ α k W ( L ∞ ,ℓ ′ ) neednot be uniformly bounded for α ∈ A since the proof of Proposition 5 only requires afinite number of terms in the amalgam norm. Remark 4.
In Proposition 5, the assumption that ( φ α ) satisfies (B1) can be re-laxed. If for all α , we have c φ α ∈ L ∞ ( R ) , with c φ α ( ξ ) ≥ on R and c φ α ( ξ ) > on T . Then ( τ α ) clearly satisfies (A1) and (A2), and we have C τ α ≤ δ − φ α k b φ α k L ∞ ( R ) C ψ , thus to account for (B2), we impose the additional hypothesis that e C Φ := sup α ∈ A δ − φ α k b φ α k L ∞ ( R ) < ∞ . For an illustration of this, see Example 4 below.
Example 1 (Convolution with the Poisson Kernel) . Let τ α ( x ) := e − α |·| ∗ ψ with ψ such that δ τ α = δ φ α δ ψ for each α . Then c τ α ( ξ ) = c φ α ( ξ ) b ψ ( ξ ) = r π αα + ξ b ψ ( ξ ) . Evidently, δ φ α = q π αα + π , and for k = 0, k c φ α ( · + 2 πk ) k L ∞ ( T ) = r π αα + (2 | k | − π . Therefore,(14) δ − φ α X k =0 k c φ α ( · + 2 πk ) k L ∞ ( T ) = X k =0 α + π α + (2 | k | − π . Note that the series on the right hand side of (14) is increasing as α increases, andmoreover each term tends to 1 as α → ∞ . Consequently, δ − φ α k c φ α k W is not boundedabove by a constant independent of α . However, δ − τ α k c τ α k W is since b ψ is compactlysupported. Indeed, if supp( b ψ ) ⊂ [ − N, N ], then (14) implies thatlim α →∞ δ − φ α N X k = − N k c φ α ( · + 2 πk ) k L ∞ ( T ) = 2 N + 1 , which implies (B2’).To verify (B3’), we use Remark 2 and simply notice that lim α →∞ α + π α + ξ = 1 uni-formly in ξ on T + 2 πk for any k ∈ {− N, . . . , N } . Example 2 (Convolution with the Gaussian Kernel) . Similar to Example 1, weobtain the same convergence results when τ α = e − α |·| ∗ ψ , letting α → ∞ . It shouldbe noted that the parameter here is the opposite as in the regular interpolators case,where the Gaussian parameter limits to 0. In both of these examples, convolutionis used to smooth out the generator with something that decays rapidly. Example 3 (Convolution with Inverse Multiquadrics) . In the vein of Examples 1and 2, consider τ α := φ α ∗ ψ where φ α ( x ) := ( x + 1) − α is the inverse multiquadricof exponent α , and we let α → ∞ . Then from [40], c φ α ( ξ ) = √ π − α Γ( α ) | ξ | α − K α − ( | ξ | ) , where K ν is the modified Bessel function of the second kind (see [1, p. 376] for theprecise definition).Since c φ α is decreasing, δ τ α = c φ α ( π ) as before. The other conditions beingeasily checked, let us consider (B3’). It suffices to check for k = − N, . . . , N that | δ − φ α c φ α ( ξ + 2 πk ) − | → ξ ∈ T . Notice that c φ α ( ξ + 2 πk ) c φ α ( π ) = | ξ + 2 πk | α − K α − ( | ξ + 2 πk | ) π α − K α − ( π ) . To consider the limit as α → ∞ , we need to know the asymptotic behavior of themodified Bessel function of the second kind with respect to its order. From [60],we find that K ν ( z ) ∼ ν − Γ( ν ) z − ν , ν → ∞ , where the notation f ( x ) ∼ g ( x ), x → ∞ means that lim x →∞ f ( x ) /g ( x ) = 1. Thus,setting ν = α − /
2, we have that c φ α ( ξ + 2 πk ) c φ α ( π ) ∼ | ξ + 2 πk | ν ν − Γ( ν ) | ξ + 2 πk | − ν π ν ν − Γ( ν ) π − ν = 1 , ν → ∞ , which yields (B3’). Example 4 (Convolution with Approximate Identities) . Finally, we may considera large class of approximate identities. These are similar in spirit to the exam-ples presented thus far, but the previous examples are not necessarily approximateidentities of the form considered here. Again make the assumption that b ψ is com-pactly supported on [ − A, A ], with N := ⌈ A ⌉ , and that X = Y . For our purposes,an approximate identity is a function φ with b φ > R , with φ, b φ ∈ L , and R R φ ( x ) dx = 1. Then set φ α ( x ) := αφ ( αx ), and we find that τ α := φ α ∗ ψ satisfies(B1)–(B4).To check that ( τ α ) α ≥ satisfies (B1)–(B4), first note that by Remark 4, (B1)is satisfied; therefore we simply verify that ( φ α ) α ≥ satisfies (B2’) and (B3’). Re-calling that c φ α ( ξ ) = b φ ( ξ/α ), it follows that the quantity δ φ α := inf ξ ∈ T | b φ ( ξ/α ) | isnon-decreasing as α increases, and hence δ − φ α ≤ δ − φ . UASI SHIFT-INVARIANT SPACES 21
Next, note that since b φ ∈ L , the inversion formula holds, and we have that | b φ ( ξ ) | ≤ k φ k L for almost every ξ ∈ R . Consequently, N X j = − N (cid:13)(cid:13)(cid:13)(cid:13)b φ (cid:18) · + 2 πjα (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( T ) ≤ (2 N + 1) k φ k L . Combining this with the previous observation about δ − φ α yields the conclusion of(B2’).To see (B3’), note that δ φ α → b φ (0) as α → ∞ , and that for any fixed j ∈{− N, . . . , N } , b φ (( ξ + 2 πj ) /α ) → b φ (0). Consequently, we have k δ − φ α b φ − k W → δ τ α / ( δ φ α δ ψ ) − Non-bandlimited ψ . Let us now consider the case when b ψ is not compactlysupported, and again X = Y . Then if τ α = φ α ∗ ψ where ( φ α ) satisfies (B1) andmodified versions of (B2’) and (B3’) where the amalgam norms therein are taken tobe W ( L ∞ , ℓ ′ ) and W ( L ∞ , ℓ ), respectively, the conclusion of Proposition 5 holdsfor ( τ α ). It should be noted that in this case, the Gaussian of Example 2 still yieldsrecovery, but convolution with the Poisson kernel does not because then the family( φ α ) α ∈ A does not satisfy the uniform bound in (B2’) when the sum is infinite (seethe discussion in Example 1).6.4. Interpolation at
X 6 = Y . It is pertinent to examine the case when supp( b ψ ) )T , and Y 6 = X . Unfortunately, recovery turns out to not be generally feasible, afact we record in the following proposition. Proposition 6.
There exist ψ with supp( b ψ ) ) T , and Y 6 = X CISs for
P W π such that there is no family ( φ α ) α ∈ A satisfying (B1) for which the interpolants I Y φ α f ∈ V ( φ α , Y ) converge in L and uniformly to f for all f ∈ V ( ψ, X ) . As of yet, we do not have enough information at our disposal to provide theproof, but we return to the matter at the end of Section 7.7.
Cardinal Functions
In this section, we analyze the special case when X = Z , in which case thespace V ( ψ ) := V ( ψ, Z ) is called the principal shift-invariant space associated withthe generator ψ – an object of extensive study in many areas of harmonic analysis,approximation theory, and functional analysis. We still make the assumptions (A1)and (A2) on ψ , and in what follows, assume that ( φ α ) α ∈ A is a one-parameter familyof generators satisfying (B1)-(B4).Cardinal interpolation arose from the penetrating work of I. J. Schoenberg onspline interpolation [58, 59], and from summability methods for the sampling seriesfound in the WKS sampling formula. Recall that if f ∈ P W π , then(15) f ( x ) = X j ∈ Z f ( j ) sinc( x − j ) , where sinc( x ) = sin( πx ) πx if x = 0, and sinc(0) = 1. Convergence of the series in(15) was later shown to be both in the sense of L ( R ) and uniform on R . But toE. T. Whittaker [65], (15) was an equation of interpolation, i.e. clearly f ( k ) = P j ∈ Z f ( j ) sinc( k − j ) for k ∈ Z since sinc( n ) = δ ,n . However, the sinc series converges slowly in the sense that sinc( x ) = O ( | x | − ).Consequently, many authors, including Schoenberg, have sought to replace sincwith another cardinal function which has the property that L ( n ) = δ ,n , n ∈ Z , butwhich decays more rapidly than sinc, hence the term summability method. Thereare of course other ways around use of the sinc kernel; for example, the generalizedsampling kernels of Butzer, Ries, and Stens [18], but we restrict our attention hereto cardinal function methods. There are many examples of such cardinal functionsand their associated decay rates [8, 14, 15, 16, 30, 31, 46, 50]. Of primary interestto us is their construction from a given function as follows.Given φ , formally define(16) c L φ ( ξ ) := 1 √ π b φ ( ξ ) X j ∈ Z b φ ( ξ + 2 πj ) . Under certain conditions (for example, if c L φ ∈ L ∩ L ) the inverse Fourier trans-form, L φ , will be a cardinal function which satisfies L φ ( k ) = δ ,k , k ∈ Z . Indeed,one needs only justify the following formal calculation: L φ ( k ) = 1 √ π X j ∈ Z Z T b φ ( ξ + 2 πj ) P m ∈ Z b φ ( ξ + 2 πm ) e i ( ξ +2 πj ) k dξ = 1 √ π Z T P j ∈ Z b φ ( ξ + 2 πj ) P m ∈ Z b φ ( ξ + 2 πm ) e iξk dξ = δ ,k . Consequently, if the family of interpolators is made from convolution with thegenerator ψ , i.e. τ α ( x ) = φ α ∗ ψ ( x ), then the Fourier transform of the cardinalfunction is d L τ α ( ξ ) = 1 √ π c φ α ( ξ ) b ψ ( ξ ) X j ∈ Z c φ α ( ξ + 2 πj ) b ψ ( ξ + 2 πj ) . Moving on to more general cardinal functions, there are a couple of naturalquestions that arise. The first is, does L φ satisfy (A1) and (A2)? Provided that φ itself does, then the answer is yes. We exhibit this in the following proposition. Proposition 7. If φ satisfies (A1) and (A2) , then L φ = √ π R R c L φ ( ξ ) e iξ · dξ is acardinal function, and moreover, L φ satisfies (A1) and (A2) .Proof. Since (A1) and (A2) hold for φ , we have b φ ( ξ ) ≥ R and δ φ >
0, thusBochner’s theorem and a routine periodization argument show that (A1) holds for L φ as well. Additionally, since b φ is nonnegative, the calculation above evaluating L φ ( k ) is valid by the monotone convergence theorem, and so L φ given by theFourier inversion formula is a cardinal function provided we have c L φ ∈ L ∩ L ,which follows from (A1).For (A2), we have X j ∈ Z k c L φ ( · + 2 πj ) k L ∞ ( T ) ≤ δ − φ X j ∈ Z k b φ ( · + 2 πj ) k L ∞ ( T ) = δ − φ k b φ k W < ∞ , UASI SHIFT-INVARIANT SPACES 23 which implies that c L φ ∈ W ( L ∞ , ℓ ). Now we can calculate C L φ by noting c L φ ≥ δ φ k b φ k W on T , which leaves us with C L φ ≤ C φ (cid:16) δ − φ k b φ ( ξ ) k L ∞ ( T ) + C φ (cid:17) < ∞ . (cid:3) Note that Proposition 7 implies that V ( L φ ) is a well-defined shift-invariant space.7.1. On Interpolation via Cardinal Functions.
Given cardinal functions con-structed previously, we now turn to their interpolation properties. Proposition 7together with Theorem 2 implies that interpolation of functions in V ( ψ ) via inter-polants in V ( L φ ) is possible. We now enumerate some of the consequences of thisfact beginning with the following lemma. For ease of notation in this section, wewrite I φ for I Z φ , representing the interpolation operator from V ( φ ) → V ( ψ ). Lemma 4.
Let I ψ be the interpolation operator associated with the generator ψ .Then if f ∈ V ( ψ ) , f = I ψ f .Proof. Note that by (5) and Theorem 2 (i) , I ψ f ∈ V ( ψ ). Moreover, I ψ f is theunique function in V ( ψ ) such that I ψ f ( k ) = f ( k ). However, evidently f ∈ V ( ψ )satisfies this relation as well; consequently I ψ f = f . (cid:3) This lemma leads us to the following proposition.
Proposition 8. If ( φ α ) satisfies (B1)–(B4) , then L φ α → L ψ both uniformly andin L ( R ) as α → ∞ .Proof. First, note that via Theorem 2 (or Lemma 2 of [45]), there exists a function f ∈ V ( ψ ) such that f ( j ) = δ ,j . For this f , I φ α f = L φ α , and f = L ψ by Lemma 4.Thus, an application of the conclusion of Theorem 4 demonstrates thatlim α →∞ I φ α f = lim α →∞ L φ α = f = L ψ uniformly and in L . (cid:3) Another, perhaps more important question, involves the form of the interpolant I φ f to a given f ∈ V ( ψ ). Theorem 2 shows that I φ f = P j ∈ Z a j φ ( · − j ) is theunique element of V ( φ ) that interpolates f at the integer lattice. However, thedefinition of the cardinal function implies that the following function interpolates f at the integers: g I φ f ( x ) := X j ∈ Z f ( j ) L φ ( x − j ) . Moreover, Lemma 1 implies that g I φ f ∈ V ( L φ ), on account of the interpolatorycondition, is the unique element of V ( L φ ) that interpolates f at Z . The followingtheorem is a consequence of the characterization of principal shift-invariant sub-spaces of L in [10], and implies that indeed g I φ f = I φ f . For completeness we givethe proof here. Theorem 6. If φ satisfies (A1) and (A2) , then V ( φ ) = V ( L φ ) . Proof.
Again, it suffices to show that F V ( φ ) = F V ( L φ ). By definition, F V ( φ ) = { P j ∈ Z c j e − ij ( · ) b φ : ( c j ) ∈ ℓ } . However, we may equivalently write the space as { Q b φ : Q | T ∈ L ( T ) , Q is 2 π –periodic } . The proof may be concluded by simplynoticing the c L φ = b φσ where σ ( ξ ) = P j ∈ Z b φ ( ξ + 2 πj ) is a continuous, 2 π –periodicfunction which is bounded above and below on T . Consequently, Q b φ = ( Q/σ ) c L φ with Q/σ a 2 π –periodic L ( T ) function, whence F V ( φ ) = F V ( L φ ). (cid:3) Consequent upon Theorem 6, the unique interpolant of f from the shift-invariantspace V ( φ ) takes the forms I φ f ( x ) = X j ∈ Z a j φ ( x − j ) = X j ∈ Z f ( j ) L φ ( x − j ) . As promised, this section concludes with the counterexample to recovery when-ever
Y 6 = X . Proof of Proposition 6.
Let Y = Z and X = Z \ { } ∪ {√ } ( Z is obviously aCIS, and the perturbation of only finitely many points in a CIS yields anotherCIS provided the resulting points are pairwise distinct). Additionally, let ψ be theGaussian kernel e −| x | , so that supp( b ψ ) = R and b ψ > R .By way of contradiction, suppose that there was a family of generators whichsatisfy (B1) such that for every f ∈ V ( ψ, X ), we have lim α →∞ I Z φ α f = f in L anduniformly, where I Z φ α f ∈ V ( φ α , Z ).First, notice that by Theorem 6, V ( φ α , Z ) = V ( L φ α , Z ). Therefore, for every f ∈ V ( ψ, X ), we have I Z φ α f = I Z L φα f = X j ∈ Z f ( j ) L φ α ( · − j )via the uniqueness of the interpolant. Consider first that \ I Z L φα ψ = X j ∈ Z ψ ( j ) e − ijξ d L φ α ( ξ ) . By the Poisson Summation Formula (which clearly holds for the Gaussian) this is X k ∈ Z b ψ ( ξ + 2 πk ) d L φ α ( ξ ) =: σ ψ ( ξ ) d L φ α ( ξ ) . Thus \ I Z L φα ψ = σ ψ d L φ α → b ψ in L , which implies that d L φ α → b ψσ ψ = c L ψ in L (since σ ψ is a 2 π –periodic function that is bounded above and below by positive constantsfor every ξ ∈ R ).Therefore, L φ α → L ψ in L ( R ). This implies that if f = ψ ( · − √ V ( ψ, X ), we have I Z L φα f → X j ∈ Z f ( j ) L ψ ( · − j )in L (this follows again by using the Poisson Summation Formula on f , which isevidently valid, and the fact that P k ∈ Z | b f ( ξ + 2 πk ) | ≤ C for ξ ∈ T ). But the righthand side above is in V ( L ψ , Z ) = V ( ψ, Z ). On the other hand, by assumption, I Z L φα f = I Z φ α f → f which is in V ( ψ, X ) \ V ( ψ, Z ), which yields a contradiction. (cid:3) UASI SHIFT-INVARIANT SPACES 25
Extensions for Cardinal Interpolation.
One of the more interesting util-ities of cardinal functions defined as in (16) is that they may still be well-definedeven when the generator φ grows. For example, if φ ( x ) := √ x + c , which is thetraditional Hardy multiquadric [33] then the cardinal function L φ is well-definedbecause b φ may be identified with a function which has an algebraic singularityat the origin and decays exponentially away from the origin [40], thus allowingthe right-hand side of (16) to be defined for every ξ ∈ R \ { } . Yet in this case, V ( L φ , Z ) = V ( φ, Z ) because the space associated with φ is not well-defined becauseof the growth of the generator. However, the decay of L φ and its Fourier transformis such that its principal shift-invariant space V ( L φ ) is indeed well-defined [14, 15].Additionally, there do exist families of cardinal functions which satisfy condi-tion (B1); as a canonical example, we consider ( L φ c ) c ∈ [1 , ∞ ) , the cardinal functionsassociated with the Hardy multiquadric mentioned above indexed by the shapeparameter c . Suppose for simplicity that supp( b ψ ) = T . Then the fact that ( L φ c )satisfies (A1), (A2), and (B2) may be surmised from [57], while (B3) is vacuousbased on the support of b ψ . Finally, (B4) follows from Proposition 2.2 of [8]. Thusthere are examples of cardinal functions which exhibit convergence by satisfyingthese conditions despite the fact that the generators they are formed from mani-festly do not. So while often the spaces V ( φ, Z ) and V ( L φ , Z ) coincide, there issometimes additional flexibility when using cardinal functions.In [46], sufficient conditions on a family of multivariate generators ( φ α ) weregiven such that cardinal interpolation from the space V ( L φ α , Z d ) (defined in theobvious manner for Z d ) is well-defined, and moreover, the interpolants of a ban-dlimited function converge to that function both in L and uniformly on R d as α → ∞ . 8. On Inverse Theorems
The conclusion of our analysis features a discussion of inverse theorems withrespect to the generators, or rather lack thereof. Indeed, it is an interesting questionwhether convergence of interpolants I Y φ α f → f for every f ∈ V ( ψ, X ) implies thatin some manner φ α → ψ . In all cases described above, the answer to this questionis negative.The first case to consider is when supp( b ψ ) = T . In this case, the fact that I Y φ α f → f for every f ∈ V ( ψ, X ) does not imply that φ α → ψ . Essentially all reg-ular interpolators of [45] are counterexamples; in particular, φ α := e − |·| α providesrecovery in V (sinc , Z ) = P W π as α → ∞ , but clearly φ α sinc in any classical(e.g. pointwise, L p , etc.) manner.For more general ψ , consider interpolation of functions in V ( ψ, Z ) via V ( φ α , Z ).Note that if I Z φ α f → f , then Theorem 6 and uniqueness of the interpolant (Theorem2) implies that I Z L φα f → f for all f as well. But Proposition 8 implies that L φ α → L ψ , whereas there are many generators for which ψ = L ψ (for example, if ψ ( x ) = e −| x | , then L ψ = ψ , a fact that can be checked via (16)), which means that wecannot also have L φ α → ψ in this case .9. Remarks
While the above analysis thoroughly explores Problem 2 in the L quasi shift-invariant space, it is natural to consider what happens in the L p setting for general p . Some things carry over in the uniform setting; for instance, under conditions (A1)and (A2), the systems { φ ( · − j ) : j ∈ Z } and { L φ ( · − j ) : j ∈ Z } are unconditionalbases for their span in L p , which we denote V p ( φ, Z ) and V p ( L φ , Z ), respectively,and moreover they are closed subspaces of L p [4]. Under the additional assumptionthat the symbol σ ( ξ ) := P j ∈ Z b φ ( ξ + 2 πk ) is in the Wiener algebra A ( T ) of 2 π –periodic functions with absolutely summable Fourier coefficients, we also have that V p ( φ, Z ) = V p ( L φ , Z ) for p ∈ [1 , σ ( ξ ) = P j ∈ Z d j e − ijξ with ( d j ) ∈ ℓ and elementary norm inequalities.Additionally, the results here involving cardinal functions extend easily to higherdimensions in the case X = Z d . However, for more general X ⊂ R d , the methodshere do not extend readily, predominantly due to the fact that Riesz bases ofexponentials are difficult to come by in higher dimensions even for straightforwarddomains (e.g. it is an open problem whether or not a Riesz basis of exponentialsexists for the Euclidean ball in R d for any d ≥ X , whichwould eliminate this difficulty, but for the interpolation method examined here, thetechniques of proof do not extend to sets X which are merely quasi-uniform, thoughthat is not to say that no such method is feasible. Acknowledgments
The first author thanks Akram Aldroubi, Alex Powell, and Ben Hayes for manyfruitful discussions involving this work. The authors also take pleasure in thankingthe anonymous referee for their valuable suggestions which greatly improved thisarticle.
References [1] M. Abramowitz and I. Stegun (Eds.),
Handbook of Mathematical Functions with Formu-las, Graphs, and Mathematical Tables , National Bureau of Standards Applied MathematicsSeries, No. 55, Courier Dover Publications, 1972.[2] A. Aldroubi, Non-uniform weighted average sampling and reconstruction in shift-invariantand wavelet spaces,
Appl. Comput. Harmon. Anal. (2002), 151-161.[3] A. Aldroubi and H. G. Feichtinger, Exact iterative reconstruction algorithm for multivariateirregularly sampled functions in spline-like spaces: the L p theory, Proc. Amer. Math. Soc. (9) (1998), 2677–2686.[4] A. Aldroubi and K. Gr¨ochenig, Nonuniform sampling and reconstruction in shift-invariantspaces,
SIAM Review , (4) (2001), 585-620.[5] N. D. Atreas, On a class of non-uniform average sampling expansions and partial reconstruc-tion in subspaces of L ( R ), Adv. Comput. Math. (1) (2012), 21-38.[6] A. Atzmon and A. Olevskii, Completeness of integer translates in function spaces on R , J.Approx. Theory (1996), 291–327.[7] B. A. Bailey, Th. Schlumprecht, and N. Sivakumar, Nonuniform sampling and recovery ofmultidimensional bandlimited functions by Gaussian radial-basis bunctions, J. Fourier Anal.Appl. (3) (2011), 519-533.[8] B. J. C. Baxter, The asymptotic cardinal function of the multiquadric φ ( r ) = ( r + c ) as c → ∞ , Comput. Math. Appl. , (12), (1992), 1-6.[9] B. J. C. Baxter and N. Sivakumar, On shifted cardinal interpolation by Gaussians and mul-tiqudrics, J. Approx. Theory (1996), 36-59.[10] C. de Boor, R. DeVore, and A. Ron, Approximation from shift-invariant subspaces of L ( R d ), Trans. Amer. Math. Soc. (2) (1994), 787-806.
UASI SHIFT-INVARIANT SPACES 27 [11] C. de Boor, R. DeVore, and A. Ron, The structure of finitely generated shift-invariant spacesin L ( R d ), J. Funct. Anal. (1) (1994), 37-78.[12] C. de Boor and A. Ron, Fourier analysis of approximation orders from principal shift-invariantspaces,
Constr. Approx. (1992), 427–462.[13] M. Bownik, The structure of shift-invariant subspaces of L ( R n ), J. Funct. Anal. (2)(2000), 282-309.[14] M. D. Buhmann, Multivariate cardinal interpolation with radial-basis functions,
Constr.Approx. (3) (1990), 225-255.[15] M. D. Buhmann, Radial basis functions: theory and implementations , Cambridge Mono-graphs on Applied and Computational Mathematics, Vol. 12. Cambridge University Press,2003.[16] M. D. Buhmann and H. Feng, On the convergence of cardinal interpolation by parameterizedradial basis functions,
J. Math. Anal. Appl. (1) (2017), 718-733.[17] M. D. Buhmann and A. Ron, Approximation orders of and approximation maps from localprincipal shift-invariant spaces,
J. Approx. Theory (1) (1995), 38-65.[18] P. L. Butzer, S. Ries, and R. L. Stens, Approximation of continuous and discontinuousfunctions by generalized sampling series, J. Approx. Theory (1987), 25-39.[19] O. Christensen, An introduction to frames and Riesz bases , Vol. 7. Boston: Birkh¨auser, 2003.[20] L. de Carli and P. Vellucci, p –Riesz bases in quasi shift invariant spaces, arXiv: 1710.00702,To Appear.[21] N. Dyn and C. Michelli, Interpolation by sums of radial functions, Numer. Math. (1990),1-9.[22] H. G. Feichtinger, Amalgam spaces and generalized harmonic analysis , Proceedings of Sym-posia in Applied Mathematics. Vol. 52. American Mathematical Society, 1997.[23] D. Freeman, E. Odell, Th. Schlumprecht, and A. Zs´ak, Unconditional structures of translatesfor L p ( R d ), Israel J. Math (2014) 189–209.[24] S. Grepstad and N. Lev, Multi-tiling and Riesz bases,
Adv. Math. (2014), 1-6.[25] K. Gr¨ochenig, C. Heil, and K. Okoudjou, Gabor analysis in weighted amalgam spaces,
Sampl.Theory Signal Image Process. (3) (2002), 225-259.[26] K. Gr¨ochenig and J. St¨ockler, Gabor Frames and Totally Positive Functions, Duke Math. J. (6) (2013), 1003-1031.[27] K. Gr¨ochenig, Jos´e Luis Romero, and J. St¨ockler, Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions,
Invent. Math. (2017)https://doi.org/10.1007/s00222-017-0760-2.[28] K. Hamm, Approximation rates for interpolation of Sobolev functions via Gaussians andallied functions,
J. Approx. Theory (2015), 101-122.[29] K. Hamm, Nonuniform sampling and recovery of bandlimited functions,
J. Math. Anal. Appl. (2) (2017), 1459-1478.[30] K. Hamm and J. Ledford, Cardinal interpolation with general multiquadrics,
Adv. Comput.Math. (5) (2016), 1149-1186.[31] K. Hamm and J. Ledford, Cardinal interpolation with general multiquadrics: convergencerates, Adv. Comput. Math.
In Press, https://doi.org/10.1007/s10444-017-9578-0.[32] T. Hangelbroek, W. R. Madych, F. Narcowich and J. Ward, Cardinal interpolation withGaussian kernels,
J. Fourier Anal. Appl. (2012), 67-86.[33] R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys.Res. (1971), 1905–1915.[34] C. Heil, An introduction to weighted Wiener amalgams, in: Wavelets and their Applications (Chennai, January 2002), M. Krishna, R. Radha, and S. Thangavelu, Eds., Allied Publishers,New Delhi 2003, 183-216.[35] K. Jetter, Riesz bounds in scattered data interpolation and L –approximation, in MultivariateApproximation: From CAGD to Wavelets, Eds. K. Jetter, M. D. Buhmann, W. Haussman,R. Schaback, and J. St¨ockler (1993), 167-177.[36] K. Jetter and J. St¨ockler, Topics in scattered data interpolation and non-uniform sampling,in Surface fitting and multiresolution methods, Eds. A. Le M´ehaute´e, C. Rabut and LLSchumaker (1997), 191-208.[37] R. Q. Jia, Shift-invariant spaces on the real line, Proc. Amer. Math. Soc. (3) (1997),785-793. [38] M. J. Johnson, On the approximation order of principal shift-invariant subspaces of L p ( R d ), J. Approx. Theory (1997), 279-319.[39] M. J. Johnson, Scattered data interpolation from principal shift-invariant spaces, J. Approx.Theory (2) (2001), 172-188.[40] D. S. Jones,
The theory of generalised functions, 2nd ed.
Cambridge University Press, Cam-bridge, 1982.[41] M. I. Kadec, The exact value of the Paley-Wiener constant,
Dokl. Adad. Nauk SSSR (1964), 1243-1254.[42] M. N. Kolountzakis, Multiple lattice tiles and Riesz bases of exponentials,
Proc. Amer. Math.Soc. (2015), 741-747.[43] G. Kozma and S. Nitzan, Combining Riesz bases,
Invent. Math. (1) (2015), 267-285.[44] G. C. Kyriazis, Approximation from shift-invariant spaces,
Constr. Approx. (1995), 141-164.[45] J. Ledford, Recovery of Paley-Wiener functions using scattered translates of regular interpo-lators, J. Approx. Theory (2013), 1–13.[46] J. Ledford, On the convergence of regular families of cardinal interpolators,
Adv. Comp.Math. (2) (2015), 357-371.[47] J. Ledford, Recovery of bivariate band-limited functions using scattered translates of thePoisson kernel, J. Approx. Theory (2015), 170-180.[48] Y. Lyubarskii, W. R. Madych, Recovery of Irregularly Sampled Band Limited Functions viaTempered Splines,
J. Funct. Anal. (1994), 201-222.[49] Y. Lyubarskii, W. R. Madych, Irregular Poisson Type Summation,
Sampl. Theory SignalImage Process. (2) (2008), 173-186.[50] W. R. Madych, Miscellaneous error bounds for multiquadric and related interpolators, Com-put. Math. Appl. (12) (1992), 121-138.[51] W. R. Madych, Summability of Lagrange type interpolation series, J. Anal. Math. (1)(2001), 207-229.[52] F. Narcowich, Recent developments in error estimates for scattered-data interpolation viaradial basis functions, Numer. Algor. (2005), 307-315.[53] F. Narcowich and J. Ward, Scattered-data interpolation on R n : error estimates for radialbasis and band-limited functions, SIAM J. Math. Anal. (1) (2004), 284–300.[54] E. Odell, B. Sari, Th. Schlumprecht, and B. Zheng, Systems formed by translates of oneelement in L p ( R ), Trans. Amer. Math. Soc. (12) (2011), 6505–6529.[55] A. Olevskii, Completeness in L ( R ) of almost integer translates, C. R. Acad. Sci. Paris (1979), 987–991.[56] B. S. Pavlov, The basis property of a system of exponentials and the condition of Mucken-houpt,
Dokl. Akad. Nauk SSSR (1979), 37-40.[57] S. D. Riemenschneider and N. Sivakumar, On the cardinal-interpolation operator associatedwith the one-dimensional multiquadric,
East J. Approx. (4) (1999), 485-514.[58] I. J. Schoenberg, Contributions to the problem of approximation of equidistant data byanalytic functions, Part A, Quart. Appl. Math. IV (1946), 45-99.[59] I. J. Schoenberg, Cardinal Spline Interpolation , Conference Board of the Mathematical Sci-ences Regional Conference Series in Applied Mathematics, Vol. 12. Society for Industrial andApplied Mathematics, Philadelphia, Pa. 1973.[60] A. Sidi, Asymptotic expansion of Mellin transforms and analogues of Watson’s lemma,
SIAMJ. Math. Anal. (1985), 896-906.[61] Th. Schlumprecht, N. Sivakumar, On the sampling and recovery of bandlimited functions viascattered translates of the Gaussian, J. Approx. Theory (2009), 128-153.[62] Q. Sun, Nonuniform Average Sampling and Reconstruction of Signals with Finite Rate ofInnovation,
SIAM J. Math. Anal. (5) (2006), 1389-1422.[63] R. M. Young, An Introduction to Nonharmonic Fourier Series , Academic Press, New York,New York, 1980.[64] H. Wendland,
Scattered Data Approximation , Cambridge Monographs on Applied and Com-putational Mathematics, Vol. 17. Cambridge University Press, 2005.[65] E. T. Whittaker, On the functions which are represented by the expansions of the interpola-tion theory,
Proc. Royal Soc. Edinburgh , Sec. A, (1915), 181–194. UASI SHIFT-INVARIANT SPACES 29
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