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.
Categories: math.CV