On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions |
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Authors: | Ferenc Móricz |
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Institution: | 1.Bolyai Institute,University of Szeged,Szeged,Hungary |
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Abstract: | We consider complex-valued functions f∈L
1(ℝ+), where ℝ+:=0,∞), and prove sufficient conditions under which the sine Fourier transform ^(f)]s\hat{f}_{s} and the cosine Fourier transform ^(f)]c\hat{f}_{c} belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere. |
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