Introduction to the Theory of Fourier Integrals |
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1+ix a(y)cos absolute convergence analytic continuation analytic function belongs to L2 bounded variation conjugate converges in mean cosh cosine transform defined e-ixw dw everywhere example f(t)cos f(u)sin finite interval follows from Theorem formally Fourier integrals Fourier kernel Fourier transforms function f(x ƒ and g gives Hankel transforms Hence f(x Hilbert transforms holds infinity inner integral integral equation Journal London Math k₁(x lemma Let f(x mean-square Mellin transforms Messenger of Math necessary and sufficient obtain Parseval's formula polynomial Proc reciprocal formula regular result follows right-hand side second mean-value theorem self-reciprocal Similarly sin ay sine transforms sinh solution strip summable Suppose tends term transform of f(x uniform convergence uniformly values Watson xt dt xy dx zeros πλ Σπί