Weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces of order $m$ in high dimensions

TL;DR

This paper introduces and analyzes weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces of order m in high dimensions, providing explicit formulas, orthonormal bases, and studying their asymptotic behavior as curvature approaches zero.

Contribution

It offers a comprehensive description of these generalized function spaces, including explicit kernels and bases, extending classical spaces to high-dimensional and entire space contexts.

Findings

Explicit formulas for reproducing kernels

Orthogonal bases constructed for the spaces

Asymptotic behavior analyzed as curvature tends to zero

Abstract

We introduce and study a generalization of the classical weighted Bergman and Dirichlet spaces on the unit ball in high dimension, the Bergman-Dirichlet spaces. Their counterparts on the whole $n$-complex space, the Bargmann-Dirichlet spaces, are also introduced and studied. Mainly, we give a complete description of the considered spaces, including orthonormal basis and the explicit formulas for their reproducing kernel functions. Moreover, we investigate their asymptotic behavior when the curvature goes to $0$.