On Stable Sampling and Interpolation in Bernstein Spaces

. We deﬁne the concepts of stable sampling set, interpolation set, uniqueness set and complete interpolation set for a quasinormed space of functions and apply these concepts to Paley-Wiener spaces and Bernstein spaces. We obtain a suﬃcient condition on a uniformly discrete set to be an interpolation set based on a lemma of convergence of series in Paley-Wiener spaces. We also obtain a result of transference, Kadec type, of the property of being a stable sampling set, from a set with this property to other uniformly discrete set, which we apply to Bernstein spaces.


Introduction
There is a large number of contributions in stable sampling and interpolation theory in Paley-Wiener and Bernstein spaces, directly and as consequence of the research in frame theory, and Riesz sequences and bases theory in L p spaces.Since the celebrated work of Claude E. Shannon ([22]), considered the father of Information Theory, research in sampling and interpolation theory (regular and irregular) has been developed in an exponential way.The contributions by A. Beurling ([4] and [5]), R. Paley and N. Wiener ( [15]), M. Plancherel and G. Pólya ( [17]), and H. J. Landau ([9]) are classical.See [7] for more details.
As said before, research in frame theory, and Riesz sequences and bases theory in L p spaces has allowed to obtain advances in stable sampling and interpolation theory.We have a particularly important example of this in 1964, when M. I. Kadec proved his celebrated theorem: Theorem 1.1 (Kadec-1/4 Theorem).Let (λ n ) n∈Z be a sequence of real numbers.Suppose that for every n ∈ Z.
Then the set of exponential functions e i λn t n∈Z is a Riesz basis in the Hilbert space L 2 ((−π, π)).
This result says, in terms of sampling and interpolation theory, that Z is a complete interpolation set (this is, both sampling and interpolation set) for the Paley-Wiener space E 2 (−π, π) , and that every L-perturbation of Z also verifies it whenever L < 1 4 (see Definition 2.2).The bound 1/4 is sharp, and Theorem 1.1 improves a previous very important result by R. Paley and N. Wiener, where the bound is 1 π 2 ([15], page 113).In 1974 S. A. Avdonin obtained a generalization of Theorem 1.1 using a certain type of mean of the values λ n 's ( [3]).
The study of sets of exponential functions is essential, and Kadec-1/4 Theorem has been generalized in several ways to L p spaces and to sequences (λ n ) n∈Z of complex numbers, in order to prove that the familiy e i λn t n∈Z , is complete (see for example [10], [19], [24] and [20]).In addition, the Riesz basis problem in the Paley-Wiener space E 2 (−π, π) has been proposed for non-exponential basis.
In this sense several important results analogous to Kadec-1/4 Theorem have been obtained for sets of sinc functions, involving the Lamb-Oseen constant (see [1] and [2]).
The aim of this paper is to establish some results of transference in irregular stable sampling and interpolation theory in Paley-Wiener spaces and Bernstein spaces.We state and prove a lemma of convergence in Paley-Wiener spaces and we obtain as consequence a sufficient condition on a uniformly discrete set to be an interpolation set.We also establish a relationship, similar to Kadec 1/4-Theorem, between the stable sampling sets of a given Bernstein space, even in a more general context, obtaining explicit sampling bounds using the Bernstein-Pesenson inequality and type Nikolskii (see [16] and [21], respectively).
We now fix some notation.We will denote by F (R n , C) (respectively, F (C n , C)) the set of the complex functions defined in R n (respectively, in C n ), by H(C n ) the set of holomorphic functions whose domain is C n , by S (R n ) the set of Schwartz functions, and by S (R n ) the space of tempered distributions.We will denote the adherence of a set A in a topological space by A. Given a Lebesgue measurable set K ⊆ R n , we denote by m n (K) its Lebesgue measure.Given A ⊆ R n , we denote the indicator function of A with respect to R n by χ A .If is a quasinorm, we denote by τ its associated topology.A function For every function f ∈ L 1 (R n ) we define the Fourier transform of f by with the usual extension to tempered distributions f ∈ S (R n ).If f is the Fourier transform of a certain tempered distribution, then we will denote by F −1 (f ), or also by f , its inverse Fourier transform.Definition 1.2 (Uniformly discrete set).Let Λ ⊆ C n be infinite countable.We say that Λ is uniformly discrete (briefly u.d.) if The constant δ(Λ) is called the separation constant of Λ.
Definition 1.3 (Uniqueness set).Let K ∈ {R, C}, and let E be a K-vector subspace of F (R n , C).Let Λ ⊆ R n be uniformly discrete.We say that Λ is a uniqueness set or complete set (briefly, US) for Definition 1.4 (Sequence space l p (Λ)).Let Λ ⊆ R n be u.d.
• We say that Λ verifies the p-Plancherel-Polya condition (briefly p-P.P.C.) for (E, ) if S is continuous, this is, if there exists a constant C > 0 such that (f (λ)) λ∈Λ p ≤ C f for each f ∈ E.
• Let A ⊆ E. Λ is said to be a p-interpolation set (in short, p-IS) for A if the restriction S| A is surjective.Given c = (c λ ) λ∈Λ ∈ l p (Λ) and f ∈ A, we say that f interpolates c (over Λ) if f (λ) = c λ for all λ ∈ Λ.
• Let A ⊆ E. We say that Λ is a p-stable sampling set (briefly, p-SS) for A if there exist constants c, C > 0, c ≤ C, such that We call C (respectively, c) an upper bound (respectively, a lower bound) of p-stable sampling for A with respect to Λ.
• We say that Λ is a p-complete interpolation set (briefly, p-CIS) for (E, ) if S is a topological isomorphism.
Remark 1.6.In the context of Definition 1.5 we observe that: (1) Λ is a uniqueness set for E if and only if S is injective.
(2) Λ is a p-CIS for (E, ) if and only if Λ is both a p-IS and a p-SS for (E, ).
If E is a vector subspace of L p (R n ), then we will refer to the p-SS, p-IS and p-CIS for (E, p ) simply as SS, IS, and CIS, respectively.
Definition 1.7.Let S ⊆ R n be a bounded set and p ∈ (0, +∞].We define which is a closed vector subspace of (L p (R n ), p ).We call (p, S)-Paley-Wiener space to the complete space (E p S , p ).Now we recall the definition of Bernstein space.
• Let σ > 0 and f ∈ H(C n ).We say that the entire function f is of exponential type at most σ if for every ε > 0 there exists • Let σ > 0 and p ∈ (0, +∞].We define the set which is a closed C-vector subspace of (L p (R n ), p ).We call (σ, p)-Bernstein space to the space (B p σ (R n ), p ), which is Banach if p ∈ [1, +∞] and is quasi-Banach if p ∈ (0, 1).
For example, the entire functions defined by z → sin(σz), z → cos(σz) are elements of B ∞ σ (R); the entire function defined by Revista Colombiana de Matemáticas belongs to B p σ (R) for all p ∈ (1, +∞].Observe that, given p ∈ (0, +∞], the Paley-Wiener subspaces and the Bernstein subspaces of (L p (R n ), p ) are invariant and isometric by translations.A very important result shows a closed relationship between both types of spaces.This result is the celebrated Paley-Wiener theorem.
Therefore Bernstein spaces are particular cases of Paley-Wiener spaces.
Now we wonder when the sampling operator is continuous, that is, when Λ verifies the p-P.P.C. for a given Paley-Wiener space (in particular, for a given Bernstein space), and the answer is: provided that Λ is uniformly discrete.This is what the following result says.
Theorem 1.10 (Plancherel-Polya inequality (see [17])).Let S ⊆ R n be a bounded set and p ∈ (0, +∞].Let Λ ⊆ R n be u.d.Then Λ verifies the p-P.P.C. for (E p S , p ), this is, there exists a constant C = C(Λ, S, p) > 0 such that Besides the constant C only depends on p, S and δ(Λ The main results of this paper are the following ones. Theorem 1.11.Let p ∈ [1, +∞), and K ⊆ R n be a bounded and Lebesgue measurable set such that its indicator function χ K is a Fourier multiplier for FL q (R n ).Let Λ ⊆ R n be u.d. and let h ∈ E 1 K be real valued or even, or both.For every λ ∈ Λ we define the translation function where δ is the Kronecker delta. Then: (2) We define the following vector subspace of E p K : The mapping ψ : (l p (Λ), p ) → (W, p ) defined by: c (4) B := {h λ } λ∈Λ is a Schauder basis for W .
We need the next definition for the following result.
For every p ∈ (0, +∞] we define The following theorem is a property of transference for stable sampling sets of certain vector subspaces of L p (R m ) similar to Kadec 1/4-Theorem (see [8], and also [25] for this and similar results).
. Suppose that the following sampling operators are continuous: given by f → (f (γ)) γ∈Γ .Let h : R m → R, h > 0, be a continuous function and let C 1 > 0 be.We define (1) There exists a function g : Γ → R, g ≥ 0, such that where | | denotes the Euclidean norm in R m . (2) for each n ∈ Z m .
(3) S < +∞, where S := Then we have: (1) If Γ is a SS for A with upper constant of sampling C > 0, and (2) If Λ is a SS for A with upper constant of sampling D > 0, and The paper is structured as follows.Section 1 contains definitions and the list of the main results.In section 2 we study the stability of SS and IS with respect to perturbations, in Bernstein spaces.Section 3 is devoted to Lemma 3.2 (the main convergence lemma) and we prove Theorem 1.11 and obtain several consequences.Section 4 contains the lemmas of continuity of immersion of Bernstein spaces in classical Bernstein spaces.In section 5 we study the transference of the property of being SS from a u.d.set to another one, and we prove Theorem 1.12 and a consequence for Bernstein spaces.In section 6 we obtain transference results of uniqueness sets between different Bernstein spaces.

Stability in sampling and interpolation in Banach spaces
Recall the next well known Banach theorem: Theorem 2.1 (Stability of surjective bounded operators).Let A : (X, X ) → (Y, Y ) be a linear, bounded and surjective operator between Banach spaces.Then there exists γ = γ(A) > 0 such that for every linear and bounded operator Definition 2.2 ( -perturbation of a set (see [14])).Let Λ, Λ ⊆ R n be u.d. and let > 0 be.Λ is said to be an -perturbation of Λ if Λ admits a representation We will need the Bernstein inequality in several dimensions, obtained by I. Pesenson.
Theorem 2.3 (Bernstein inequality (see [16]) The following result give sense to the word stable in stable sampling and interpolation theory.
Theorem 2.4 (Principle of stability with respect to perturbations).Let Λ ⊆ R n be uniformly discrete and p ∈ [1, +∞].Let σ > 0. If Λ is a SS (respectively, IS, CIS) for B p σ (R n ), there exists > 0 such that every -perturbation of Λ is a SS (respectively, IS, CIS) for B p σ (R n ).
Proof.Our proof is similar to that by A. Olevskii and A. Ulanovskii for the case p = 2, n = 1 (see [14]).
First we take p ∈ [1, +∞).We claim that there exists a constant Using the Bernstein inequality (Theorem 2.3), the mean value theorem and Theorem 1.10, we have: where ξ λ belongs to the segment with extreme points λ, λ + ε λ , for all λ ∈ Λ.
In the penultimate inequality we have applied Theorem 1.10 to each partial derivative ∂f ∂xj ∈ B p σ (R n ), with j ∈ {1, ..., n}.Observe that the set Γ Λ := {ξ λ : λ ∈ Λ} is a uniformly discrete set (in fact it is an ε-perturbation of Λ), and therefore we can apply Theorem 1.10.
Then as Λ is uniformly discrete, we have: Hence δ(Λ) < 2 ε + ξ λ − ξ λ , and so As we are assuming that ε < δ(Λ) 4 , then we obtain that Notice that, in particular, our claim shows that (1) Suppose that Λ is an IS for B p σ (R n ).This is, S Λ is surjective.Let γ = γ(S Λ ) > 0 given by stability theorem for bounded surjective operators.Taking ε sufficiently small we have S Λ − S Λ < γ.So S Λ is surjective by stability theorem; this is, Λ is an IS for B p σ (R n ).
(2) Suppose that Λ is a SS for B p σ (R n ).Then by Theorem 1.10 there exists Let f ∈ B p σ (R n ).Then: On the other hand, as Λ is a SS for B p σ (R n ), then we have: Taking > 0 enough small we have that 1 − R • n Therefore Λ is a SS for B p σ (R n ).For the case p = +∞ we have that there exists a constant D = D (σ, δ(Λ)) > 0 such that and the proof is completely analogous.

Lemma of Convergence.
In this section we prove a result of convergence of series in Paley-Wiener spaces which allows to make sure the convergence of series under certain conditions.First we need the following auxiliary and well known result: Lemma 3.1.Let r ∈ (0, +∞] and S ⊆ R n .Let g ∈ E r S be even.Then: (1) The real and imaginary parts of g and their Fourier transform, Re(g), Re(g), Im g, Img, are even.
(2) Re(g)(t) ∈ R and Im g(t) ∈ R for each t ∈ R n .
(3) Re(g), Im g ∈ E r S .Lemma 3.2 (Lemma of Convergence).Let p ∈ [1, +∞), and K ⊆ R n be a bounded and Lebesgue measurable set such that its indicator function χ K is a Fourier multiplier for FL q (R n ).Let Λ ⊆ R n be a uniformly discrete set and let h ∈ E 1 K be real valued.For every λ ∈ Λ we define the function h λ := τ λ h : R n −→ R by h λ (x) := (τ λ h) (x) = h(x − λ) for all x ∈ R n .Then there exists a constant D > 0 such that In particular, for each c = (c λ ) λ∈Λ ∈ l p (Λ) we have that g c := λ∈Λ1 c λ • h λ ∈ L p (R n ), and thus g c ∈ E p K .In addition, the constant D only depends on p, h 1 , δ(Λ) and on K. Remark 3.3.Observe that Lemma 3.2 is also true for even functions h ∈ E 1 K .This is an immediate consequence of applying the lemma to the real and imaginary parts of h, by Lemma 3.1.In addition, note that Proof.If h = 0, the result is obvious.Suppose that h is not the function identically zero.
Since χ K is a Fourier multiplier for FL q (R n ), then Volumen 56, Número 2, Año 2022 where C q,h := C q • h 1 > 0. In addition, using the Plancherel-Polya inequality, Theorem 1.10, we have: So that |< g c , ϕ >| ≤ D • c p • ϕ q , and this is true for each ϕ ∈ S (R n ).Hence Now we prove Theorem 1.11.

Proof of Theorem 1.11
Proof.
(1) We know by Lemma 3.2 of convergence that the series That is, converges uniformly in R n to g c , and thus also converges pointwise.So that for each µ ∈ Λ. Hence g c interpolates c and g c = λ∈Λ g c (λ) h λ .
(2) Obviously ψ is surjective, by definition of W .Let c = (c λ ) λ∈Λ ∈ l p (Λ), and define h := λ∈Λ c λ h λ .Then we have: In addition, by Lemma 3.2 of convergence there exists a constant D 2 > 0 independent of c such that Then, Thus ψ is continuous.We will prove that ψ is an isomorphism showing that there exists a constant D 1 > 0 such that We shew in the proof of the first item that and therefore g c = λ∈Λ g c (λ) • h λ ∈ E p K .Then: where we have used the Plancherel-Polya inequality (Theorem 1.10). ( By definition of W we have that there exists c = (c λ ) λ∈Λ ∈ l p (Λ) such that g = λ∈Λ c λ • h λ .Since ψ is injective, then c is unique, and besides we have So that c µ = g(µ) for all µ ∈ Λ and hence (4) It has been proved in the previous item.(2) h(γ) = 0 for every γ ∈ Γ \ {0}.
Then Λ is an IS for (B p υ , p ).In fact, for each c = (c λ ) λ∈Λ ∈ l p (Λ) we have that Proof.By Paley-Wiener Theorem 1.9 we have that

and the indicator function of [−υ, υ]
n is a Fourier multiplier for FL q (R n ), where q is the conjugate exponent of p. Then by Theorem 1.11 we obtain the result.Lemma 4.2.Let p ∈ (0, +∞).
(1) Let Ω ⊆ C be a connected and open set.Then |f | p is subharmonic for every f ∈ H(Ω).
(3) Let n ∈ N, n > 1.Let r ∈ {1, ..., n − 1}, and let a = (a 1 , ..., a r ) ∈ R r .Then: (a) The function In fact, we have: Proof.Two first items are well known (for the first item see [18], page 336, Theorem 17.3, and for the second one see [21], Chapter 6).In the third item the only non obvious thing is the continuity of ψ a .
First, we shall prove it for r = 1 (step from n to n − 1).
C := C π, p , a ∈ R. ψ a has the form: The step from n to n − r is obtained by reiteration.(1) The inclusion is continuous.That is: There exists C σ, p > 0 such that We may take C σ, p := C π, p • σ π 1/p = 2σ pπ 3 (e pπ − 1) , as said in the statement of Lemma 4.2.
(1) This is a consequence of the second item of Lemma 4.2 using a change of variable.By the previous lemma we know that the inclusion where C π, p = 2 pπ 2 (e pπ − 1) (see [21], Chapter 6).
Let us consider the linear operators defined by f → P 1 (f ) : C → C given by z → P 1 (f )(z) := f π σ z , and defined by f → P 2 (f ) : C → C given by z → P 2 (f )(z) := f σ π z .Both operators are continuous since and Since ϕ σ = P 2 • ϕ π • P 1 , then ϕ σ is continuous, and in fact we have (2) The proof is analogous to the one of the third item of Lemma 4.2.
(3) This is a consequence of the previous item using induction over the dimension n.For n = 1 the result is true by the first item.Assume that the result is true for n = k ∈ Z, k > 0. We will show that the result is true for n = k + 1.
We will show that f ≤ C (f (λ)) λ∈Λ p for every f ∈ A.
Let f ∈ A. There exists a sequence (g n ) n∈Z + in A which converges to f in X.As S is continuous, then (S (g n )) n∈Z + converges to S (f ) in l p (Λ), and consequently lim n→+∞ (g n (λ)) λ∈Λ p = lim n→+∞ S (g n ) p = S (f ) p = (f (λ)) λ∈Λ p .
Let n ∈ Z + .Since g n ∈ A, then our assumption gives us Taking limits in this inequality when n → +∞ we finally obtain what concludes the proof.
Let us prove now Theorem 1.12.

Proof of Theorem 1.12
Proof.Take {|< ∇F (x), a >|} ≤ max where the first inequality is consequence of the Cauchy-Schwarz inequality.
Therefore for each n ∈ Z m we have: Let F ∈ A. We may distinguish two cases: (1) Case I: p ∈ (0, +∞).Then we have: (2) Case II: p = +∞.Then we have: So that we have proved our claim.Now we are in conditions to prove the items of the theorem.
(1) Assume that Γ is a SS for A with upper constant of sampling C > 0, and S < 1 C•C2 .We will prove that Λ is a SS for A with upper constant of > 0, which is also independent of F .Then: ) Assume that Λ is a SS for A with upper constant of sampling D > 0, and S < 1 D•C2 .We will prove that Γ is a SS for A with upper constant of sampling > 0, which is also independent of F .Then: Conclusion: Γ is a SS for A with upper constant of sampling and, by Lemma 5.2, it is also a SS for A with the same upper constant of sampling.
Now we obtain two consequences of Theorem 1.12.The first one is the particular case Γ = Z m and the second one is the application of Theorem 1.12 to Bernstein spaces.
where the second inequality is consequence of Theorem 2.3, and the last inequality is for p ∈ [1, +∞).
Theorem 1.12 concludes the proof.