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

Rachid Conte; Ousmane Toure; Mamadouba Toure; Aboubakary Diakhaby

doi:doi:10.11648/j.ijtam.20261203.12

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

Rachid Conte, Ousmane Toure, Mamadouba Toure, Aboubakary Diakhaby

Published in International Journal of Theoretical and Applied Mathematics (Volume 12, Issue 3)

Received: 28 May 2026 Accepted: 10 June 2026 Published: 12 August 2026

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Abstract

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.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 12, Issue 3)
DOI	10.11648/j.ijtam.20261203.12
Page(s)	69-73
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Regulated Functions, Fourier Series, Generalized Orlicz Sequence Spaces, Modular Spaces, N-function, △₂-condition

References

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APA Style

Conte, R., Toure, O., Toure, M., Diakhaby, A. (2026). Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces. International Journal of Theoretical and Applied Mathematics, 12(3), 69-73. https://doi.org/10.11648/j.ijtam.20261203.12

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ACS Style

Conte, R.; Toure, O.; Toure, M.; Diakhaby, A. Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces. Int. J. Theor. Appl. Math. 2026, 12(3), 69-73. doi: 10.11648/j.ijtam.20261203.12

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AMA Style

Conte R, Toure O, Toure M, Diakhaby A. Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces. Int J Theor Appl Math. 2026;12(3):69-73. doi: 10.11648/j.ijtam.20261203.12

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@article{10.11648/j.ijtam.20261203.12,
  author = {Rachid Conte and Ousmane Toure and Mamadouba Toure and Aboubakary Diakhaby},
  title = {Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {12},
  number = {3},
  pages = {69-73},
  doi = {10.11648/j.ijtam.20261203.12},
  url = {https://doi.org/10.11648/j.ijtam.20261203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20261203.12},
  abstract = {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.},
 year = {2026}
}

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TY  - JOUR
T1  - Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces
AU  - Rachid Conte
AU  - Ousmane Toure
AU  - Mamadouba Toure
AU  - Aboubakary Diakhaby
Y1  - 2026/08/12
PY  - 2026
N1  - https://doi.org/10.11648/j.ijtam.20261203.12
DO  - 10.11648/j.ijtam.20261203.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 69
EP  - 73
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20261203.12
AB  - 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.
VL  - 12
IS  - 3
ER  -

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Author Information

Rachid Conte

Department of Basic and Applied Sciences, Higher Institute of Mines and Geology of Boke, Boke, Guinea

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Ousmane Toure

Mathematics Department, Gamal Abdel Nasser University, Conakry, Guinea

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Mamadouba Toure

Mathematics Department, Gamal Abdel Nasser University, Conakry, Guinea

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Aboubakary Diakhaby

West African Institute of Mathematics, Gamal Abdel Nasser University, Conakry, Guinea

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APA Style

Conte, R., Toure, O., Toure, M., Diakhaby, A. (2026). Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces. International Journal of Theoretical and Applied Mathematics, 12(3), 69-73. https://doi.org/10.11648/j.ijtam.20261203.12

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ACS Style

Conte, R.; Toure, O.; Toure, M.; Diakhaby, A. Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces. Int. J. Theor. Appl. Math. 2026, 12(3), 69-73. doi: 10.11648/j.ijtam.20261203.12

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AMA Style

Conte R, Toure O, Toure M, Diakhaby A. Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces. Int J Theor Appl Math. 2026;12(3):69-73. doi: 10.11648/j.ijtam.20261203.12

Copy | Download

@article{10.11648/j.ijtam.20261203.12,
  author = {Rachid Conte and Ousmane Toure and Mamadouba Toure and Aboubakary Diakhaby},
  title = {Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {12},
  number = {3},
  pages = {69-73},
  doi = {10.11648/j.ijtam.20261203.12},
  url = {https://doi.org/10.11648/j.ijtam.20261203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20261203.12},
  abstract = {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.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Convergence of Fourier Series of Regulated Functions in Generalized Orlicz Sequence Spaces
AU  - Rachid Conte
AU  - Ousmane Toure
AU  - Mamadouba Toure
AU  - Aboubakary Diakhaby
Y1  - 2026/08/12
PY  - 2026
N1  - https://doi.org/10.11648/j.ijtam.20261203.12
DO  - 10.11648/j.ijtam.20261203.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 69
EP  - 73
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20261203.12
AB  - 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.
VL  - 12
IS  - 3
ER  -

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