The Approximation of Functions of Several Variables with Bounded p-Fluctuation by Polynomials in the Walsh System

This paper presents direct and inverse theorems concerning the approximation of functions of several variables with bounded p-fluctuation using Walsh polynomials. These theorems provide estimates for the best approximation of such functions by polynomials in the norm of the space under consideration. The paper investigates the properties of the Walsh system, which includes piecewise constant functions, and builds on earlier work on trigonometric and multiplicative systems. The results are theoretical and have potential applications in such areas as coding theory, digital signal processing, pattern recognition, and probability theory.