Polyanalytic Gaussian Radial Basis Function Kernel and Itô-Hermite Polynomials
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918161765367808 |
|---|---|
| author | De Bie, Hendrik De Martino, Antonino Diki, Kamal |
| author_facet | De Bie, Hendrik De Martino, Antonino Diki, Kamal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14639 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polyanalytic Gaussian Radial Basis Function Kernel and Itô-Hermite Polynomials De Bie, Hendrik De Martino, Antonino Diki, Kamal Functional Analysis 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. |
| title | Polyanalytic Gaussian Radial Basis Function Kernel and Itô-Hermite Polynomials |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2510.14639 |