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

Abstract

We consider the systems of rational functions $\{{\Phi }_n(z)\},n\in \mathbb{Z}$, defined by fixed set points $\mathbf{a}:=\{a_k{\}}_{k=0}^{\infty },(\mathrm{Im}{a}_k>0)$, $\mathbf{b}:=\{b_k{\}}_{k=1}^{\infty },(\mathrm{Im}{b}_k<0)$ 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.