Sampling in spaces of entire functions of exponential type in ☆,☆☆
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 -functions whose Fourier transform is supported in the interval . 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 whose restrictions to the surface have
The Sobolev space and the Fourier transform
We consider the fractional operator defined following [16], [10] as where is an integer, whose domain is the set of for which the limit exists in . Then is a closed operator on , and its domain contains the Schwartz space , see [10, Thm. (3.15)]. When , the operator has an inverse given by convolution with a locally integrable homogeneous function. We denote such convolution operator by . 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 are complete, for .
A representation theorem for
In this section we prove a representation theorem for functions in , for . We denote by the 1-dimensional Euclidean Fourier transform, that is, for , We also write . Recall that extends to a surjective isomorphism where and the inverse is defined as We recall that if and has compact support, then f extends to an entire function F and
Theorem 4.1 Let ,
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 . 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 in is sampling for if and only if ([24], [25]) and that the behavior of the sampling constants as is
Sampling in
Before proving Theorem 1.6, we study a few properties of . 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 the Fock spaces that will appear in this section are defined for negative λ in and that, by definition, .
We use both the notation and to denote the 1-dimensional Euclidean Fourier transform of . Let such that ,
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 and some sufficient conditions for sampling sequences for the Fock space as in [15].
Moreover, in this paper we essentially dealt with the Hilbert case
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Cited by (0)
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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.