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Coefficient Condition forL1-Convergence of Walsh–Fourier Series

✍ Scribed by S. Fridli

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
195 KB
Volume
210
Category
Article
ISSN
0022-247X

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✦ Synopsis

In this paper we give a condition with respect to Walsh᎐Fourier coefficients that implies the L -convergence of the corresponding Walsh᎐Fourier series. We show 1 that the L -convergence class induced by this condition contains each one of the 1 previously known convergence classes as a proper subset. We also show that our condition implies not only the L -convergence but also the convergence in the 1 dyadic Hardy norm if the function represented by the series belongs to the dyadic Hardy space. ᮊ 1997 Academic Press MAIN RESULT Throughout this paper ‫,ގ‬ ‫,ސ‬ ‫ޒ‬ will stand for the set of natural numbers, positive integers, and real numbers, respectively. Let r denote the kth k Rademacher function, i.e., q1 if 0Fx-1r2, r x s Ž . 0 ½ y1 if 1r2Fx-1 periodic by 1, and r x s r 2 k x 0 F x-1, g ‫ސ‬ .

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