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

2015/07/03 by S. O. Chaichenko, Chaichenko, S. O.
Mathematics · #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1507.00994

openalex publication_date 2015/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the systems of rational functions \Φ_n(z)\, ~n ∈ ℤ, defined by fixed set points \bf a:=\a_k\_k=0^∞, ~ (\mathop\rm Im a_k>0), \bf b:=\b_k\_k=1^∞, ~ (\mathop\rm Im b_k<0) and is orthonormal on the real axis ℝ. We have obtained the compact form of analogue of Dirichlet kernels of these systems on the real axis ℝ. Using obtained representation we investigate the problems of convergence in the spaces L_p(ℝ),~ p> 1, and pointwise convergence of Fourier series on the systems \Φ_n(t)\,~ n ∈ 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.

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