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Smoothness conditions and integrability theorems on bounded Vilenkin groups

  • Walter R. Bloom (a1) and John J. F. Fournier (a2)

Abstract

Various criteria, in terms of forward differences and related operations on coefficients, are shown to imply that certain series on bounded Vilenkin groups represent integrable functions. These results include analogues of known integrability theorems for trigonometric series. The method of proof is to pass from the given series to a derived series, and to deduce the integrability of the original series from smoothness properties of the latter.

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References

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[1]Baiarstanova, A. S., ‘On a class of series with monotone coefficients’, Izv. Vysh. Uchebn. Zaved. Mat. 27 (12) (1983), 37; translated in Soviet Math. (Izv. VUZ) 27 (12) (1983), 3–7.
[2]Balashov, L. A., ‘On series with respect to the Walsh system with monotone coefficients’, Sibirsk. Mat. Zh. 12 (1971), 2539; translated in Siberian Math. J. 12 (1971), 18–28.
[3]Bloom, Walter R. and Fournier, John J. F., ‘Generalized Lipschitz spaces on Vilenkin groups’, Math. Nachr. 132 (1987), 6780.
[4]Bray, William O. and Stanojević, Vera B., ‘On the integrability of complex trigonometric series’, Proc. Amer. Math. Soc. 93 (1985), 5158.
[5]Edwards, R. E., Fourier series, a modern introduction (Holt, Rinehart and Winston, New York, 1967).
[6]Fomin, G. A., ‘A class of trigonometric series’, Mat. Zametki 23 (1978), 213222; translated in Math. Notes 23 (1978), 117–123.
[7]Fournier, John J. F. and Self, W., ‘Some sufficient conditions for uniform convergence of Fourier series’, J. Math. Anal. Appl. 126 (1987), 355374.
[8]Onneweer, C. W., ‘On the definition of dyadic differantiation’, Applicable Anal. 9 (1979), 267278.
[9]Vilenkin, N. Ya., ‘On a class of complete orthonormal systems’, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), 363400; translated in Amer. Math. Soc. Transl. 28 (1963), 1–35.
[10]Yano, Shigeki, ‘On Walsh-Fourier series’, Tôhoku Math. J. 3 (1951), 223242.
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Smoothness conditions and integrability theorems on bounded Vilenkin groups

  • Walter R. Bloom (a1) and John J. F. Fournier (a2)

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