A construction of certain nonlinear approximating families

Abstract

1. In discussing the nonlinear approximations of A. G. Vitushkin [2], G. G. Lorentz [1] points out a discrepancy between Vitushkin's definition of the degree of an algebraic polynomial in several variables and the usual definition of algebraic geometry. Lorentz observes that Vitushkin's reliance on a theorem of algebraic geometry makes this deviation unjustified. This paper is written to substantiate Lorentz and begin to fill the resulting gap in Vitushkin's theory. Vitushkin considers, among others, approximating families F, in a real normed linear space L, of the form