Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On double cosine, sine, and Walsh series with monotone coefficients

Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 109 (1990), 417-425
MSC: Primary 42B05; Secondary 42A32
MathSciNet review: 1010803
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We extend from one-dimensional to two-dimensional series the results by Hardy and Littlewood [6] on the $ {L^r}$-integrability of the sum $ f$ and the results by Stechkin [10] on the $ {L^1}$-integrability of the maximum partial sum $ {M^*}$ in the case of cosine and sine series with monotone coefficients. Among others, we prove that the $ {L^r}$-integrability of $ f$ and $ {M^*}$ is essentially equivalent for $ r > 1$ in the two-dimensional setting, too. Simultaneously, we extend our earlier results in [7] from one-dimensional to two-dimensional Walsh series.

References [Enhancements On Off] (What's this?)

  • [1] G. Alexits, Convergence problems of orthogonal series, Pergamon Press, New York-Oxford-Paris, 1961. MR 0218827 (36:1911)
  • [2] M. I. D'yachenko, On the convergence of double trigonometric series and Fourier series with monotone coefficients, Math. USSR-Sb. 57 (1987), pp. 57-75. MR 830095 (87i:42025)
  • [3] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), pp. 372-414. MR 0032833 (11:352b)
  • [4] G. H. Hardy, Notes on some points in the integral calculus (LXIV), Messenger for Math. 57 (1928), pp. 12-16.
  • [5] G. H. Hardy and J. E. Littlewood, Elementary theorems concerning power series with positive coefficients and moment constants of positive functions, J. Reine Angew. Math. 157 (1927), pp. 141-158.
  • [6] -, Some new properties of Fourier constants, J. London Math. Soc. 6 (1931), pp. 3-9.
  • [7] F. Móricz, On Walsh series with coefficients tending monotonically to zero, Acta. Math. Acad. Sci. Hungar. 38 (1981), pp. 183-189. MR 634579 (82m:42019)
  • [8] -, On the integrability of double cosine and sine series. I, J. Math. Anal. Appl. (submitted).
  • [9] F. Móricz, F. Schipp, and W. Wade, On the integrability of double Walsh series with special coefficients, Michigan Math. J. (submitted).
  • [10] S. B. Stechkin, On power series and trigonometric series with monotone coefficients (in Russian), Uspekhi Mat. Nauk 18(1) (1963), pp. 173-180. MR 0146583 (26:4105)
  • [11] A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 1959.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42B05, 42A32

Retrieve articles in all journals with MSC: 42B05, 42A32

Additional Information

Keywords: Rectangular partial sum, maximum partial sum, pointwise convergence and convergence in $ {L^r}$-norm in Pringsheim's sense, Hardy's inequalities for double integrals and double sequences
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society