On a generalization of the system of functions of Rademacher and Walsh

Abstract

In the paper, we propose generalizations of the Rademacher and Walsh functions. They are based on $????_2$-representation of real numbers generalizing classic binary representation of numbers $????\in[0;1]$. We study integral properties of these functions. In particular, we prove that generalized Rademacher functions form an orthogonal system of functions. For any generalized Walsh function, its analytic expression is found.