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A theorem for Fourier coefficients of a function of class \(L^ p\). (English) Zbl 0724.42006

The author proves the following theorem. Let \(\{a_ n\}\) be a positive null sequence such that \(n^{-\beta}a_ n\) is monotonically decreasing for some integer \(\beta\geq 0\), \(A_ n=(1/n)\sum^{n}_{k=1}a_ k,\) \(1\leq p<\infty,\) and \(-1<\alpha p<p-1.\) Then \(\sum^{\infty}_{n=1}a_ n \cos nx\) is the Fourier series of a function in L(p,\(\alpha\)) if and only if \(\sum^{\infty}_{n=1}A_ n \cos nx\) is the Fourier series of another function in L(p,\(\alpha\)). We remind the reader that a function f is said to belong to the class L(p,\(\alpha\)) if \(\int^{\pi}_{0}| f(x)|^ p(\sin x)^{\alpha p}dx<\infty\).
Reviewer: F.Móricz (Szeged)

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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