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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On the degree of approximation of a function by the partial sums of its Fourier series


Author: Elaine Cohen
Journal: Trans. Amer. Math. Soc. 235 (1978), 35-74
MSC: Primary 42A08
MathSciNet review: 0461004
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Abstract: When f is a $ 2\pi $ periodic function with rth order fractional derivative, $ r \geqslant 0$, of p-bounded variation, Golubov has obtained estimates of the degree of approximation of f, in the $ {L^q}$ norm, $ q > p$, by the partial sums of its Fourier series. Here we consider the analogous problem for functions whose fractional derivatives are of $ \Phi $-bounded variation and obtain estimates of the degree of approximation in an Orlicz space norm. In a similar manner we shall extend various results that he obtained on degree of approximation in the sup norm.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0461004-9
Keywords: Orlicz class, Orlicz space, $ {\Delta _2}$-condition, $ \Delta '$-condition, $ \Phi $-modulus of continuity, $ \Phi $-variation, fractional integral, fractional derivative, degree of approximation
Article copyright: © Copyright 1978 American Mathematical Society