Pansions and the theory of Fourier transforms
Jacob Korevaar · Transactions of the American Mathematical Society · 1959
is called the conjugate Fourier transform off. The famous inversion theorem for the Fourier transformation says that if g= Tf then f= Tg. Unfortunately this basic result is true only under rather severe restrictions, for example if both f and Tf are in L1 [3, p. 51 ]. Other well-known rules of restricted validity are the following: TDf= {ix}Tf, T{x}f= iDTf, TE(D)f = { E(ix) } Tf, T{ E(x) }f = E(iD) Tf, T(fi * f2) = Tf1 * Tf2. Here D stands for differentiation, E(x) = EanXn denotes an entire function, and fl*f2 stands for the convolution