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absolute convergence analytic continuation analytic function argument belongs to L2 bounded variation conjugate converges in mean converges uniformly cosine transform deduce defined differential equations everywhere example exists Fc(x finite interval follows from Theorem formally Fourier integrals Fourier kernel Fourier series Fourier transforms function f(x G. H. HARDY gives Hankel transforms Hardy Hence Hilbert transforms holds ia+oo Iff(x infinity inner integral integral equation integral function inversion formulae inversion is justified Journal London Math lemma Let f(x lim f limit line of integration Mellin transforms Messenger of Math necessary and sufficient obtain oo,oo Oxford Parseval formula Parseval's formula polynomial positive Proc Quart reciprocal formula regular result follows right-hand side second mean-value theorem self-reciprocal functions similar Similarly sine transforms solution strip summability Suppose tends term Theorem 14 theory of Fourier Titchmarsh transform of f(x uniform convergence values Watson zeros