Estimates of the norms of some cosine and sine series

Yıl 2023, Cilt: 6 Sayı: 3, 142 - 150, 15.09.2023

Jorge Bustamante

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

Öz

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

Anahtar Kelimeler

Fourier series, Dirichlet kernel, cosine and sine sums.

Kaynakça

• 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).