Sampling in spaces of entire functions of exponential type in Cn+1,☆☆

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Abstract

In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose on the entire functions, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary bU of the Siegel upper half-space U and it is fundamental that bU can be identified with the Heisenberg group Hn. We consider entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restriction to bU are square integrable with respect to the Haar measure on Hn. For these functions we prove a version of the Whittaker–Kotelnikov–Shannon Theorem. Instrumental in our work are spaces of entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restrictions to bU belong to some homogeneous Sobolev space on Hn. For these spaces, using the group Fourier transform on Hn, we prove a Paley–Wiener type theorem and a Plancherel–Pólya type inequality.

Section snippets

Introduction and statement of the main results

The classical Paley–Wiener theorem characterizes the entire functions of exponential type a in the complex plane whose restriction to the real line is square integrable as the space of L2-functions whose Fourier transform is supported in the interval [a,a]. For such functions perhaps the most far reaching result is the Whittaker–Kotelnikov–Shannon sampling theorem. These results have been extended to several variables for functions in Cn whose restrictions to the surface {Imz=0}=Rn have

The Sobolev space W˙s,p and the Fourier transform

We consider the fractional operator Δs/2 defined following [16], [10] asΔs/2φ=limε01Γ(ks2)εrks21erΔΔkφdr, where k>s/2 is an integer, whose domain is the set of φLp(Hn) for which the limit exists in Lp. Then Δs/2 is a closed operator on Lp, 1<p< and its domain contains the Schwartz space S, see [10, Thm. (3.15)]. When 0<s<2n+2, the operator Δs/2 has an inverse given by convolution with a locally integrable homogeneous function. We denote such convolution operator by Is. The following

The Plancherel–Pólya inequality

In this section we prove our first results. We begin with a Plancherel–Pólya type inequality adapted to the Siegel half-space. This result implies in particular that the spaces PWas are complete, for 0s<n+1.

A representation theorem for PWas

In this section we prove a representation theorem for functions in PWas, for s[0,n+1). We denote by F the 1-dimensional Euclidean Fourier transform, that is, for fL1(R),Ff(ξ)=Rf(x)eixξdx. We also write Ff=fˆ. Recall that F extends to a surjective isomorphism F:L2(R)L2(R) wherefL2(R)2=12πFfL2(R)2 and the inverse F1 is defined asF1f(x)=12πRf(ξ)eiξxdξ. We recall that if fL2 and fˆ has compact support, then f extends to an entire function F andF(F(+iy))(λ)=fˆ(λ)eyλ.

Theorem 4.1

Let FPWas, 0s<n+1

Sampling in the Fock space

In this section we prove a result, Theorem 5.3, that it may be considered as folklore. We consider the 1-dimensional case and make explicit the dependence on λ of the sampling constant in the case of square lattices for the Fock space Fλ(C). However, we believe that the result is not completely obvious and it is key for our Theorem 1.6. We recall that a square lattice Lb in C is sampling for Fa(C) if and only if b>a ([24], [25]) and that the behavior of the sampling constants as ba+ is

Sampling in PWa

Before proving Theorem 1.6, we study a few properties of PWan. In particular we present some elements and produce an explicit orthonormal basis of such space. We also remark that because of the Fourier transform characterization of PWan the Fock spaces Fλ that will appear in this section are defined for negative λ in [a,0) and that, by definition, Fλ=F|λ|.

We use both the notation Fg and gˆ to denote the 1-dimensional Euclidean Fourier transform of gL2(R). Let gL2(R) such that suppgˆ[a,0],

Final remarks and open questions

We believe that the spaces we introduced are worth investigating and arise quite naturally in our multi-dimensional setting.

The present work leaves some open questions. First of all, it should be proved a more general version of Theorem 1.6 by combining the characterization of sampling sequences for the 1-dimensional Paley–Wiener space PWa and some sufficient conditions for sampling sequences for the Fock space F(Cn) as in [15].

Moreover, in this paper we essentially dealt with the Hilbert case

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    The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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    The first author is partially supported by the 2020 INdAM/GNAMPA grant Alla frontiera tra l'analisi complessa in più variabili e l'analisi armonica. The second and third authors are partially supported by the 2020 INdAM/GNAMPA grant Fractional Laplacians and subLaplacians on Lie groups and trees.

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