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Estimates of the norms of some cosine and sine series

Year 2023, Volume: 6 Issue: 3, 142 - 150, 15.09.2023

Jorge Bustamante

https://doi.org/10.33205/cma.1345440

Abstract

In the work we estimate the $\mathbb{L}^1$ norms of some special cosine and sine series used in studying fractional integrals.

Keywords

Fourier series Dirichlet kernel cosine and sine sums.

References

  • N. K. Bari: Trigonometric Series, Moscow (in Russian, 1961).
  • R. G. Bartle: The Elements of Integration, John Wiley and Sons, Inc., New York-London-Sydney (1966).
  • A. S. Belov: On the unimprovability of some theorems on the convergence in the mean of trigonometric series, J. Math. Sci. (N.Y.), 250(3) (2020), 404–418.
  • G. Brown, K. Y. Wang and D. C. Wilson: Positivity of some basic cosine sums, Math. Proc. Cambridge Philos. Soc., 114(3) (1993), 383–391.
  • P. L. Butzer, R. J. Nessel: Fourier Analysis and Approximation, New York-Basel (1971).
  • P. L. Butzer, U.Westphal: An access to fractional differentiation via fractional difference quotients, Fractional Calculus and Its Applications, Lecture Notes in Mathematics, 457 (1975), 116–145.
  • J. W. Garrett, Cˇ . V. Stanojevic´: Necessary and sufficient conditions for $\mathbb{L}^1$ convergence of trigonometric series, Proc. Amer. Math. Soc., 60 (1976), 68–71.
  • J.W. Garrett, Cˇ . V. Stanojevic´: On $\mathbb{L}^1$ convergence of certain cosine sums, Proc. Amer.Math. Soc., 54 (1976), 101–105.
  • M. Izumi, Sh. Izumi: On some trigonometrical polynomials, Math. Scand., 21 (1967), 38–44.
  • I. P. Natanson: Constructive Function Theory. Vol I, Frederick Ungar Publ., New York (1964).
  • B. Szal: On L-convergence of trigonometric series, J. Math. Anal. Appl., 373(2) (2011), 449–463.
  • Z. Tomovski: Convergence and integrability for some classes of trigonometric series, Dissertationes Math (Rozprawy Mat.), 420 (2003), 65 pp.
  • W. H. Young: On a certain series of Fourier, Proc. London Math. Soc., 11 (1913), 357–366.
  • A. Zygmund: Trigonometric series, Third Edition, Vol I and II combined, Cambridge Mathematical Library (2002).

Year 2023, Volume: 6 Issue: 3, 142 - 150, 15.09.2023

Jorge Bustamante

https://doi.org/10.33205/cma.1345440

Abstract

References

  • N. K. Bari: Trigonometric Series, Moscow (in Russian, 1961).
  • R. G. Bartle: The Elements of Integration, John Wiley and Sons, Inc., New York-London-Sydney (1966).
  • A. S. Belov: On the unimprovability of some theorems on the convergence in the mean of trigonometric series, J. Math. Sci. (N.Y.), 250(3) (2020), 404–418.
  • G. Brown, K. Y. Wang and D. C. Wilson: Positivity of some basic cosine sums, Math. Proc. Cambridge Philos. Soc., 114(3) (1993), 383–391.
  • P. L. Butzer, R. J. Nessel: Fourier Analysis and Approximation, New York-Basel (1971).
  • P. L. Butzer, U.Westphal: An access to fractional differentiation via fractional difference quotients, Fractional Calculus and Its Applications, Lecture Notes in Mathematics, 457 (1975), 116–145.
  • J. W. Garrett, Cˇ . V. Stanojevic´: Necessary and sufficient conditions for $\mathbb{L}^1$ convergence of trigonometric series, Proc. Amer. Math. Soc., 60 (1976), 68–71.
  • J.W. Garrett, Cˇ . V. Stanojevic´: On $\mathbb{L}^1$ convergence of certain cosine sums, Proc. Amer.Math. Soc., 54 (1976), 101–105.
  • M. Izumi, Sh. Izumi: On some trigonometrical polynomials, Math. Scand., 21 (1967), 38–44.
  • I. P. Natanson: Constructive Function Theory. Vol I, Frederick Ungar Publ., New York (1964).
  • B. Szal: On L-convergence of trigonometric series, J. Math. Anal. Appl., 373(2) (2011), 449–463.
  • Z. Tomovski: Convergence and integrability for some classes of trigonometric series, Dissertationes Math (Rozprawy Mat.), 420 (2003), 65 pp.
  • W. H. Young: On a certain series of Fourier, Proc. London Math. Soc., 11 (1913), 357–366.
  • A. Zygmund: Trigonometric series, Third Edition, Vol I and II combined, Cambridge Mathematical Library (2002).
There are 14 citations in total.

Details

Primary Language English
Subjects Lie Groups, Harmonic and Fourier Analysis
Journal Section Articles
Authors

Jorge Bustamante 0000-0003-2856-6738

Early Pub Date August 18, 2023
Publication Date September 15, 2023
Published in Issue Year 2023 Volume: 6 Issue: 3

Cite

APA Bustamante, J. (2023). Estimates of the norms of some cosine and sine series. Constructive Mathematical Analysis, 6(3), 142-150. https://doi.org/10.33205/cma.1345440
AMA Bustamante J. Estimates of the norms of some cosine and sine series. CMA. September 2023;6(3):142-150. doi:10.33205/cma.1345440
Chicago Bustamante, Jorge. “Estimates of the Norms of Some Cosine and Sine Series”. Constructive Mathematical Analysis 6, no. 3 (September 2023): 142-50. https://doi.org/10.33205/cma.1345440.
EndNote Bustamante J (September 1, 2023) Estimates of the norms of some cosine and sine series. Constructive Mathematical Analysis 6 3 142–150.
IEEE J. Bustamante, “Estimates of the norms of some cosine and sine series”, CMA, vol. 6, no. 3, pp. 142–150, 2023, doi: 10.33205/cma.1345440.
ISNAD Bustamante, Jorge. “Estimates of the Norms of Some Cosine and Sine Series”. Constructive Mathematical Analysis 6/3 (September 2023), 142-150. https://doi.org/10.33205/cma.1345440.
JAMA Bustamante J. Estimates of the norms of some cosine and sine series. CMA. 2023;6:142–150.
MLA Bustamante, Jorge. “Estimates of the Norms of Some Cosine and Sine Series”. Constructive Mathematical Analysis, vol. 6, no. 3, 2023, pp. 142-50, doi:10.33205/cma.1345440.
Vancouver Bustamante J. Estimates of the norms of some cosine and sine series. CMA. 2023;6(3):142-50.
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