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

Abstract

The systems of rational functions {Φ_n(z)}, n ∈ ℤ; that are orthonormalized on the real axis ℝ and are defined by the fixed set of points a := {a_k}_k = 0^∞, (Im a_k > 0) and b := {b_k}_k = 1^∞, (Im b_k < 0); are considered. Some analogs of the Dirichlet kernels of the systems {Φ_n(t)}, n ∈ ℤ; on the real axis ℝ are given in a compact form, and the convergence in the spaces L_p(ℝ); p > 1; and the pointwise convergence of Fourier series on the systems {Φ_n(t)}, n ∈ ℤ; are studied under the certain restrictons on the sequences of poles of these systems. Some analogs of the classical Jordan–Dirichlet and Dini–Lipschitz criteria of convergence of Fourier series in a trigonometric system are constructed.