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On the Mittag Leffler Bargmann (MLB) transform

Natanael Alpay, Kamal Diki

TL;DR

This work develops a Segal-Bargmann type transform for Mittag-Leffler Fock spaces, establishing a unitary map from $L^2(\mathbb{R})$ onto the Mittag-Leffler Fock space and revealing a precise algebraic link to the Fourier transform through an intertwining operator. The MLB transform is built from a generating kernel derived from Hermite functions, maps Hermite bases to Mittag-Leffler monomials, and admits an explicit inverse; it provides a robust framework for connecting MLQ-analysis with classical harmonic analysis. The paper also introduces Caputo-derivative based creation/annihilation operators, derives commutation relations, and presents explicit cases along with a conjectured general form for integer orders. Finally, it extends the theory to quaternionic slice-hyperholomorphic settings, defining a quaternionic Mittag-Leffler Fock space and a corresponding QB transform, thereby broadening the functional-analytic and operator-theoretic toolkit for hypercomplex analysis.

Abstract

We introduce the Segal-Bargmann transform associated to the Mittag Leffler Fock space and study how it will be connected to the Fourier transform. We will discuss also the counterpart of the creation and annihilation operator in this setting using the Caputo and Liouville operators. Finally, we give an extension of these results to the case of quaternions, in particular in the slice hyperholomorphic setting.

On the Mittag Leffler Bargmann (MLB) transform

TL;DR

This work develops a Segal-Bargmann type transform for Mittag-Leffler Fock spaces, establishing a unitary map from onto the Mittag-Leffler Fock space and revealing a precise algebraic link to the Fourier transform through an intertwining operator. The MLB transform is built from a generating kernel derived from Hermite functions, maps Hermite bases to Mittag-Leffler monomials, and admits an explicit inverse; it provides a robust framework for connecting MLQ-analysis with classical harmonic analysis. The paper also introduces Caputo-derivative based creation/annihilation operators, derives commutation relations, and presents explicit cases along with a conjectured general form for integer orders. Finally, it extends the theory to quaternionic slice-hyperholomorphic settings, defining a quaternionic Mittag-Leffler Fock space and a corresponding QB transform, thereby broadening the functional-analytic and operator-theoretic toolkit for hypercomplex analysis.

Abstract

We introduce the Segal-Bargmann transform associated to the Mittag Leffler Fock space and study how it will be connected to the Fourier transform. We will discuss also the counterpart of the creation and annihilation operator in this setting using the Caputo and Liouville operators. Finally, we give an extension of these results to the case of quaternions, in particular in the slice hyperholomorphic setting.
Paper Structure (6 sections, 13 theorems, 83 equations)

This paper contains 6 sections, 13 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Mittag Leffler Bargmann transform
  4. MLB and Fourier transform
  5. Creation and annihilation operators
  6. Quaternionic MLB transform

Key Result

Proposition 3.4

For any $z,w\in\mathbb{C}$, it holds that In particular, we have also

Theorems & Definitions (44)

  • Definition 2.1: Fock space
  • Definition 2.2: Inner product on $L^2(\mathbb{R})$ and $\mathcal{F}$
  • Definition 2.3: Bargmann transform
  • Definition 2.4: MLF space
  • Remark 2.5
  • Definition 2.6: Caputo fractional derivative
  • Definition 2.7: Caputo fractional derivative in ML spaces
  • Definition 3.1: ML-Bargmann kernel
  • Definition 3.2: MLB transform
  • Remark 3.3
  • ...and 34 more