Notes on the Overconvergence of Fourier Series and Hadamard–Ostrowski Gaps

Abstract

This paper examines the relationship between the overconvergence of Fourier series and the existence of Hadamard–Ostrowski gaps. Ostrowski’s result on the overconvergence of power series serves as a motivating factor for obtaining a natural generalization: the overconvergence of Fourier series. The connection between Hadamard–Ostrowski gaps and the overconvergence of Fourier series is clarified by applying the Hadamard three-circle theorem and the theory of orthogonal polynomials. Our main result is obtained by applying the Hadamard three-circle theorem.

Keywords

• Fourier series
• overconvergence
• orthogonal polynomials
• Hadamard–Ostrowski gaps