About Absolute Convergence of Fourier Series of Almost Periodic Functions

Farkhodzhon Makhmadshevich Talbakov

doi:doi:10.11648/j.pamj.20241303.11

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About Absolute Convergence of Fourier Series of Almost Periodic Functions

Farkhodzhon Makhmadshevich Talbakov^*

Published in Pure and Applied Mathematics Journal (Volume 13, Issue 3)

Received: 18 February 2024 Accepted: 12 March 2024 Published: 2 July 2024

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Abstract

The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x₁, x_2,...x_p), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x₁=a₁ t, x₂=a₂ t,..., x_p=a_pt, where a₁, a₂, ..., a_p - some real, non-zero numbers. Considering the value of the function f(x₁, x_2,...x_p) on this line, he obtains a function of one variable φ(t) = f(a₁ t, a₂ t,...a_p t) and proves that this function is almost periodic. With some choice of numbers a₁, a₂, ..., a_p - it may happen that this function is periodic. However, if the numbers a₁, a₂, ..., a_p are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a₁, a₂, ..., a_p associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λ_k→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.

Published in	Pure and Applied Mathematics Journal (Volume 13, Issue 3)
DOI	10.11648/j.pamj.20241303.11
Page(s)	36-43
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), 2024. Published by Science Publishing Group

Keywords

Almost Periodic Bohr Functions, Fourier Series, Spectrum Functions, Fourier Coefficients, Trigonometric Polynomials, Best Uniform Approximation, Limit Point at Zero, Laplace Transform

References

[1]	Bari N. K. Trigonometric series M.: Fizmatgiz. -1961. - 936 p.
[2]	Besicovitch A. S. Almost periodic functions. Cambridge. – 1932. - 180 р.
[3]	Bohr G. Almost-periodic functions. M.: LIBROCOM. - 2009. - 128 p.
[4]	Sigmund A. Trigonometric series, Cambridge Univ. Press, 1959. 615 p.
[5]	Levitan B. M. Almost periodic functions. -M.: GITTL, 1953, 396 p.
[6]	Bredikhina E. A. On the absolute convergence of Fourier series of almost periodic functions. - DAN USSR, 1956, v. 111, no. 6, p. 1163-1166.
[7]	Bredikhina E. A. Some estimates of deviations of partial sums of Fourier series from almost periodic functions. -Math. collection, 1960, t. 50(92), No. 3, pp. 369-382.
[8]	Musielak J. O. bezwzglednejz bieznosciczeregow Fourier apewnych funcjiprawieokresowich.-Bull. Acad. Polon. Sci. Ci., 1957, v. 3, № 5, pp. 9-17.
[9]	Kuptsov N. P. On the absolute and uniform convergence of Fourier series of almost periodic functions. -Math. collection, 1956, t. 40(82), no. 2, p. 157-178.
[10]	Pritula Ya. G. About the absolute significance of the Fourier series of periodic functions. - Visnik Lviv. un-tu, 1971, v. 137, no. 5, p. 72-80.
[11]	Jafarov A. S., Mamedov G. A. On the absolute convergence of Fourier series of almost periodic Besicovitch functions.-Izv. AN Azerbaijan. SSR, 1983, No. 5, p. 8-13.
[12]	Khasanov Yu. Kh. On the absolute convergence of Fourier series of almost periodic functions. -Math. notes, 2013, vol. 94, no. 5, p. 745-756.
[13]	Khasanov Yu. Kh., Talbakov F. M. An analogue of S. N. Bernstein's theorem on the best approximation of almost periodic functions. DAN RT. – 2016. – vol. 59. – No. 1-2. – pp. 11-18.
[14]	Khasanov Yu. Kh., Talbakov F. M. On the absolute convergence of Fourier series of almost periodic Bezikovich functions. DAN RT. – 2018. – vol. 61. - No. 1-2. – pp. 813-821.
[15]	Talbakov F. M. About absolute convergence of Fourier series of almost periodic functions in the uniform metric - DAN RT, 2020, vol. 63, No. 5-6, p. 289-293.
[16]	Talbakov F. M. On the absolute convergence of double Fourier series of almost-periodic functions in a uniform metric. News of universities. Mathematics. - 2023. – No. 4. – pp. 65-75.

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Talbakov, F. M. (2024). About Absolute Convergence of Fourier Series of Almost Periodic Functions. Pure and Applied Mathematics Journal, 13(3), 36-43. https://doi.org/10.11648/j.pamj.20241303.11

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Talbakov, F. M. About Absolute Convergence of Fourier Series of Almost Periodic Functions. Pure Appl. Math. J. 2024, 13(3), 36-43. doi: 10.11648/j.pamj.20241303.11

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AMA Style

Talbakov FM. About Absolute Convergence of Fourier Series of Almost Periodic Functions. Pure Appl Math J. 2024;13(3):36-43. doi: 10.11648/j.pamj.20241303.11

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@article{10.11648/j.pamj.20241303.11,
  author = {Farkhodzhon Makhmadshevich Talbakov},
  title = {About Absolute Convergence of Fourier Series of Almost Periodic Functions
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {13},
  number = {3},
  pages = {36-43},
  doi = {10.11648/j.pamj.20241303.11},
  url = {https://doi.org/10.11648/j.pamj.20241303.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241303.11},
  abstract = {The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x1, x2,...xp), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x1=a1 t, x2=a2 t,..., xp=apt, where a1, a2, ..., ap - some real, non-zero numbers. Considering the value of the function f(x1, x2,...xp) on this line, he obtains a function of one variable φ(t) = f(a1 t, a2 t,...ap t) and proves that this function is almost periodic. With some choice of numbers a1, a2, ..., ap - it may happen that this function is periodic. However, if the numbers a1, a2, ..., ap are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a1, a2, ..., ap associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λk→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.
},
 year = {2024}
}

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TY  - JOUR
T1  - About Absolute Convergence of Fourier Series of Almost Periodic Functions

AU  - Farkhodzhon Makhmadshevich Talbakov
Y1  - 2024/07/02
PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241303.11
DO  - 10.11648/j.pamj.20241303.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 36
EP  - 43
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241303.11
AB  - The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x1, x2,...xp), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x1=a1 t, x2=a2 t,..., xp=apt, where a1, a2, ..., ap - some real, non-zero numbers. Considering the value of the function f(x1, x2,...xp) on this line, he obtains a function of one variable φ(t) = f(a1 t, a2 t,...ap t) and proves that this function is almost periodic. With some choice of numbers a1, a2, ..., ap - it may happen that this function is periodic. However, if the numbers a1, a2, ..., ap are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a1, a2, ..., ap associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λk→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.

VL  - 13
IS  - 3
ER  -

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Farkhodzhon Makhmadshevich Talbakov

Department of Geometry and Higher Mathematics, Tajik State Pedagogical University Named After Sadriddin, Aenya, Dushanbe, Tajikistan

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APA Style

Talbakov, F. M. (2024). About Absolute Convergence of Fourier Series of Almost Periodic Functions. Pure and Applied Mathematics Journal, 13(3), 36-43. https://doi.org/10.11648/j.pamj.20241303.11

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ACS Style

Talbakov, F. M. About Absolute Convergence of Fourier Series of Almost Periodic Functions. Pure Appl. Math. J. 2024, 13(3), 36-43. doi: 10.11648/j.pamj.20241303.11

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AMA Style

Talbakov FM. About Absolute Convergence of Fourier Series of Almost Periodic Functions. Pure Appl Math J. 2024;13(3):36-43. doi: 10.11648/j.pamj.20241303.11

Copy | Download

@article{10.11648/j.pamj.20241303.11,
  author = {Farkhodzhon Makhmadshevich Talbakov},
  title = {About Absolute Convergence of Fourier Series of Almost Periodic Functions
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {13},
  number = {3},
  pages = {36-43},
  doi = {10.11648/j.pamj.20241303.11},
  url = {https://doi.org/10.11648/j.pamj.20241303.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241303.11},
  abstract = {The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x1, x2,...xp), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x1=a1 t, x2=a2 t,..., xp=apt, where a1, a2, ..., ap - some real, non-zero numbers. Considering the value of the function f(x1, x2,...xp) on this line, he obtains a function of one variable φ(t) = f(a1 t, a2 t,...ap t) and proves that this function is almost periodic. With some choice of numbers a1, a2, ..., ap - it may happen that this function is periodic. However, if the numbers a1, a2, ..., ap are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a1, a2, ..., ap associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λk→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - About Absolute Convergence of Fourier Series of Almost Periodic Functions

AU  - Farkhodzhon Makhmadshevich Talbakov
Y1  - 2024/07/02
PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241303.11
DO  - 10.11648/j.pamj.20241303.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 36
EP  - 43
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241303.11
AB  - The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x1, x2,...xp), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x1=a1 t, x2=a2 t,..., xp=apt, where a1, a2, ..., ap - some real, non-zero numbers. Considering the value of the function f(x1, x2,...xp) on this line, he obtains a function of one variable φ(t) = f(a1 t, a2 t,...ap t) and proves that this function is almost periodic. With some choice of numbers a1, a2, ..., ap - it may happen that this function is periodic. However, if the numbers a1, a2, ..., ap are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a1, a2, ..., ap associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λk→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.

VL  - 13
IS  - 3
ER  -

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