Fourier Transforms

complex FS of f on −L ≤ ≤ L. This function can be written asH ∑n−c nexp i nL~ f .From the earlier theory on Fourier series, we know that the series H converges to f at points of continuity and that the coefficients c n are given byLc n 12L−L H L ∑ n− f exp −i nLd L f nLf nLexp i nL.The FT f is band-limited and we can relate this to H by introducing the functionG such thatto giveG 1 when || L 0 when || L. f ~ H G .Thus f is the product of two functions and can be considered to be the product of twotransforms, so that ft could be written in convolution form. It is slightly easier, andequivalent, to employ the direct inversion formula to obtainft ~ 12 H G e−itd .−Now substitute for the function G andthenforH, ftft ~ 12 12 12L~ ∑n−L−L∑n− L−LH e−itdL∑n−f nL−Lf nLsinn − Ltn − Ltf nLexp i nLLexp i nL− t de −it dand which can be made exact by careful justification. This enables an approximation to be19