Walsh Functions and Their Generalizations

Abstract

Consider the function defined on the half open unit interval [0, 1) by

$${{r}_{0}}(x) = \left\{ {\begin{array}{*{20}{c}} 1 & {for x \in [0, 1/2),} \\ { - 1} & {for x \in [1/2,1).} \\ \end{array} } \right.$$

Extend it to the real line by priodicity of period 1 and set r _k (x) = r₀(2^kx)for k = 0,1, … and real x. the functions r _k (x)are called the Rademacher functions.