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Polyanalytic Gaussian Radial Basis Function Kernel and Itô-Hermite Polynomials

Hendrik De Bie, Antonino De Martino, Kamal Diki

TL;DR

This work constructs a polyanalytic extension of the Gaussian RBF kernel by analyzing the convolution of modulated Hermite functions and applying the Zaremba–Bergman formula. The resulting kernels K_{RBF,N}(z,w)=e^{- rac{1}{4}(z-ar{w})^2} L^{1}_{N-1}( rac{|z-w|^2}{2}) reproduce Hilbert spaces that are isomorphic to polyanalytic Fock spaces and admit a Landau-type operator description. The authors develop an operator framework linking true polyanalytic RBF spaces to polyanalytic Fock spaces, construct polyanalytic Weyl operators, and derive applications via the Christoffel–Darboux formula and Mehler kernels. The approach yields explicit bases, reproducing kernels, and unitarity properties that facilitate time-frequency and quantum-harmonic analysis in the polyanalytic setting, with implications for kernel methods and complex analysis. Overall, the paper provides a coherent analytic bridge between polyanalytic function theory, reproducing-kernel spaces, and operator methods in a Gaussian RBF context.

Abstract

We introduce a polyanalytic extension of the Gaussian radial basis function (RBF) kernel by computing the action of the convolution operator on normalized Hermite functions. In particular, using the Zaremba-Bergman formula we derive an explicit closed form for this new reproducing kernel function. We then establish an isomorphism relating the reproducing kernel Hilbert space induced by the polyanalytic Gaussian RBF kernel with the corresponding polyanalytic Fock space. Moreover, we provide a characterization of polyanalytic Gaussian RBF spaces in terms of a Landau-type operator. In addition, we investigate the polyanalytic counterpart of the Weyl operator, which leads to applications involving the Christoffel-Darboux formula for Hermite polynomials and Mehler's kernel. Finally, we discuss the analogue of the Weyl operator in the context of the polyanalytic Gaussian RBF setting.

Polyanalytic Gaussian Radial Basis Function Kernel and Itô-Hermite Polynomials

TL;DR

This work constructs a polyanalytic extension of the Gaussian RBF kernel by analyzing the convolution of modulated Hermite functions and applying the Zaremba–Bergman formula. The resulting kernels K_{RBF,N}(z,w)=e^{- rac{1}{4}(z-ar{w})^2} L^{1}_{N-1}( rac{|z-w|^2}{2}) reproduce Hilbert spaces that are isomorphic to polyanalytic Fock spaces and admit a Landau-type operator description. The authors develop an operator framework linking true polyanalytic RBF spaces to polyanalytic Fock spaces, construct polyanalytic Weyl operators, and derive applications via the Christoffel–Darboux formula and Mehler kernels. The approach yields explicit bases, reproducing kernels, and unitarity properties that facilitate time-frequency and quantum-harmonic analysis in the polyanalytic setting, with implications for kernel methods and complex analysis. Overall, the paper provides a coherent analytic bridge between polyanalytic function theory, reproducing-kernel spaces, and operator methods in a Gaussian RBF context.

Abstract

We introduce a polyanalytic extension of the Gaussian radial basis function (RBF) kernel by computing the action of the convolution operator on normalized Hermite functions. In particular, using the Zaremba-Bergman formula we derive an explicit closed form for this new reproducing kernel function. We then establish an isomorphism relating the reproducing kernel Hilbert space induced by the polyanalytic Gaussian RBF kernel with the corresponding polyanalytic Fock space. Moreover, we provide a characterization of polyanalytic Gaussian RBF spaces in terms of a Landau-type operator. In addition, we investigate the polyanalytic counterpart of the Weyl operator, which leads to applications involving the Christoffel-Darboux formula for Hermite polynomials and Mehler's kernel. Finally, we discuss the analogue of the Weyl operator in the context of the polyanalytic Gaussian RBF setting.
Paper Structure (17 sections, 50 theorems, 302 equations, 1 figure, 1 table)

This paper contains 17 sections, 50 theorems, 302 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analytic case
  4. Polyanalytic case
  5. Polyanalytic and true polyanalytic Fock spaces
  6. Complex Itô-Hermite polynomials
  7. Convolution operator and Gaussian RBF kernel: analytic setting
  8. Polyanalytic Gaussian RBF kernels
  9. An operator approach to polyanalytic Gaussian RBF kernels
  10. Weyl operators for polyanalytic functions
  11. Polyanalytic Weyl operators and time-frequency shifts
  12. Applications: Christoffel-Darboux formula and Mehler's kernel
  13. The Weyl operator on a polyanalytic Gaussian RBF space
  14. Appendix
  15. Results with a generic parameter
  16. ...and 2 more sections

Key Result

Theorem 2.6

The Segal-Bargmann transform $\mathcal{B}$ defines a unitary operator that maps the Hilbert space $L^2(\mathbb{R})$ onto the Fock space $\mathcal{F}(\mathbb{C})$. Furthermore, $\mathcal{B}$ transforms the orthonormal basis $(\psi_k)_{k\in\mathbb{N}_0}$ onto an orthonormal basis of $\mathcal{F}(\math

Figures (1)

  • Figure 1: Flowchart of proposed approaches for constructing the polyanalytic Gaussian RBF kernels.

Theorems & Definitions (161)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7: Weyl operator
  • Remark 2.8
  • Definition 2.9: Gaussian RBF kernel
  • Remark 2.10
  • ...and 151 more