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

DOI:
https://doi.org/10.1090/S0002-9939-1990-1010803-4

MathSciNet review:
1010803

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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 -integrability of the sum and the results by Stechkin [10] on the -integrability of the maximum partial sum in the case of cosine and sine series with monotone coefficients. Among others, we prove that the -integrability of and is essentially equivalent for in the two-dimensional setting, too. Simultaneously, we extend our earlier results in [7] from one-dimensional to two-dimensional Walsh series.

**[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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1010803-4

Keywords:
Rectangular partial sum,
maximum partial sum,
pointwise convergence and convergence in -norm in Pringsheim's sense,
Hardy's inequalities for double integrals and double sequences

Article copyright:
© Copyright 1990
American Mathematical Society