Convergence of Fourier series on the system of rational functions on the real axis

TL;DR

This paper studies the convergence properties of Fourier series formed by rational functions orthonormal on the real axis, providing new convergence theorems analogous to classical Fourier series results.

Contribution

It derives a compact form of Dirichlet-like kernels for rational function systems and establishes convergence theorems under specific pole restrictions.

Findings

Established pointwise convergence of Fourier series on rational systems.

Proved convergence in Lp spaces for p > 1.

Derived analogues of classical Fourier convergence theorems.

Abstract

We consider the systems of rational functions $\{\Phi_n(z)\}, ~n \in \mathbb{Z}$, defined by fixed set points ${\bf a}:=\{a_k\}_{k=0}^{\infty}, ~ (\mathop{\rm Im} a_k>0)$, ${\bf b}:=\{b_k\}_{k=1}^{\infty}, ~ (\mathop{\rm Im} b_k<0)$ and is orthonormal on the real axis $\mathbb{R}.$ We have obtained the compact form of analogue of Dirichlet kernels of these systems on the real axis $\mathbb{R}.$ Using obtained representation we investigate the problems of convergence in the spaces $L_p(\mathbb{R}),~ p> 1,$ and pointwise convergence of Fourier series on the systems $\{\Phi_n(t)\},~ n \in \mathbb{Z},$ provided that the sequences of poles of these systems satisfies certain restrictions. We have proved statements that are analogues of the classical Theorems of Jordan-Dirichlet and Dini-Lipschitz of convergence of Fourier series on the trigonometric system.