Approximation by linear combinations of continuous functions with restricted coefficients

Abstract

We prove necessary and sufficient conditions for linear combinations of given functions K k ϵ C[ a, b] ( k = 1, 2, …) to be dense in C[ a, b], C 0[ a, b] respectively, where the coefficients satisfy bound constraints. Our general approach to such problems is based on a new method of functional analysis concerning a relationship between approximations in normed linear spaces with generalized restrictions, defined by seminorms, and certain corresponding properties of bounded linear functionals on these spaces. In the special case of approximation by Müntz polynomials with restricted coefficients some known results and sharpened versions of these can be deduced from our general theorems. Finally, an application to another type of function K k is obtained, where K k ( t) = φ( tλ k −1), λ k → ∞, with φ being certain analytic functions.