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Generalized Fock space and fractional derivatives with Applications to Uniqueness of Sampling and Interpolation Sets

Natanael Alpay, Paula Cerejeiras, Uwe Kähler

TL;DR

The paper develops a generalized Fock space framework tied to Gelfond-Leontiev derivatives, including Dunkl-type operators, and constructs a modified Bargmann transform to relate L^2-space problems to analytic function spaces. It establishes Beurling-density-based density theorems for sampling and interpolation within these fractional Fock spaces and derives lattice conditions guaranteeing frames for associated integral transforms. A fractional analogue of the Weierstrass-σ function is introduced to facilitate the density analysis, with applications to Gabor-type frames and Dunkl-driven transforms. The results provide a pathway to frame construction and coherent-state representations in generalized derivative settings with potential applications in quantum mechanics and signal processing.

Abstract

In this paper we introduce a Fock space related to derivatives of Gelfond-Leontiev type, a class of derivatives which includes many classic examples like fractional derivatives or Dunkl operators. For this space we establish a modified Bargmann transform as well as density theorems for sampling and interpolation. These density theorems allow us to establish lattice conditions for the construction of frames arising from integral transforms which are linked by the modified Bargmann transform with the Fock space.

Generalized Fock space and fractional derivatives with Applications to Uniqueness of Sampling and Interpolation Sets

TL;DR

The paper develops a generalized Fock space framework tied to Gelfond-Leontiev derivatives, including Dunkl-type operators, and constructs a modified Bargmann transform to relate L^2-space problems to analytic function spaces. It establishes Beurling-density-based density theorems for sampling and interpolation within these fractional Fock spaces and derives lattice conditions guaranteeing frames for associated integral transforms. A fractional analogue of the Weierstrass-σ function is introduced to facilitate the density analysis, with applications to Gabor-type frames and Dunkl-driven transforms. The results provide a pathway to frame construction and coherent-state representations in generalized derivative settings with potential applications in quantum mechanics and signal processing.

Abstract

In this paper we introduce a Fock space related to derivatives of Gelfond-Leontiev type, a class of derivatives which includes many classic examples like fractional derivatives or Dunkl operators. For this space we establish a modified Bargmann transform as well as density theorems for sampling and interpolation. These density theorems allow us to establish lattice conditions for the construction of frames arising from integral transforms which are linked by the modified Bargmann transform with the Fock space.
Paper Structure (13 sections, 17 theorems, 187 equations)

This paper contains 13 sections, 17 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized fractional derivatives
  4. Dunkl operators
  5. Fractional Fock space
  6. Inner product
  7. Generalized Bargmann transform
  8. Density Theorems for Sampling and Interpolations
  9. Necessary Conditions for Interpolation and Sampling
  10. A Fractional Analogue of the Weierstrass-$\sigma$ function
  11. Proof of main theorem for the quasi-periodic case
  12. Proofs of main theorems for the subharmonic case
  13. Applications

Key Result

Lemma 2.1

If the multiplicity function $\kappa$ is such that $\cap_{j} \ker T_j = \mathbb{C}$ then it exists a unique positive linear isomorphism $V_{\kappa} : \Pi \to \Pi,$ denoted as intertwining operator, which satisfies

Theorems & Definitions (49)

  • Definition 2.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 2.1: Roesler
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • ...and 39 more