DOI: 10.11648/j.ijtam.20261203.12 ISSN: 2575-5080

Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces

Rachid Conte, Ousmane Toure, Mamadouba Toure, Aboubakary Diakhaby
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The study of Fourier coefficients in function spaces is a classical topic in harmonic analysis, with foundational results by Hardy, Littlewood, and Zygmund for Lebesgue and Orlicz spaces. This paper investigates the convergence of Fourier series of regulated functions, which are uniform limits of step functions, within the framework of generalized Orlicz sequence spaces. Regulated functions form a broad class that includes continuous, monotone, and piecewise continuous functions, making them natural candidates for studying Fourier series at points of discontinuity. The purpose of this work is to establish a continuity result for the Fourier coefficient operators acting from an Orlicz space of functions into a generalized Orlicz sequence space. The methodology relies on a modular approach using a non-decreasing sequence of N-functions, combined with a uniform Delta-2 condition and a controlled growth condition on the sequence. A classical Hausdorff-Young estimate is used to relate the decay of Fourier coefficients to the modular of the function. The main result (Theorem 4.1) states that the maps sending a regulated function to its sequences of cosine and sine Fourier coefficients are continuous linear operators under the stated hypotheses. Examples including the sawtooth wave and piecewise constant functions illustrate the theory. These results extend classical Hardy?Littlewood type theorems to a broader class of sequence spaces. The findings contribute to the theory of modular spaces and Fourier analysis, with potential applications to partial differential equations and approximation theory.

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