Some Simple Unitary Transformations

Abstract

(a) If f(x) e S then Uf(x) = g(x) e S. (b) f I g(x) 12dx = f If(x) I2dx. 00 00 (c) U has an inverse; i.e. if g(x) e S then g(x) = Uf(x) for some f(x) e S. (d) U is a linear transformation; i.e. U(kifi(x) + k2f2(x)) = k1Uf1(x) + k2Uf2(x) for complex constants k1 and k2. A well known example is the Fourier transform, g(x) = (2r) eix tf(t) dt, 00 which is unitary if S is taken to be the class L2 of Lebesgue integrable functions and if (c) and (d) are taken in the almost everywhere sense. The purpose of the present note is to analyze some transformations of simple form which are unitary over the class S of continuous functions such that 00