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 ${\bf a}:=\{a_k\}_{k=0}^{\infty}, ~ (\mathop{\rm Im} a_k>0)$, ${\bf b}:=\{b_k\}_{k=1}^{\infty}, ~ (\mathop{\rm Im} b_k<0)$ 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.