Jackson-Steklov Approximation Theory in Weighted Sobolev Spaces with Operations Research Applications

Abstract

This paper studies the theoretical framework that brings together the Jackson-Steklov theory of approximation in weighted Sobolev spaces and optimization techniques used in Operations Research (OR). In this work, several new direct Jackson-type theorems are established, and a detailed analysis of Steklov eigenvalues in weighted  spaces is carried out, along with error bounds and conditioning results for polynomial approximations. The results of this paper are intended for use in Operations Research, where stochastic programming with heavy-tailed distributions, PDE constrained optimization with irregular domains, and regularized inverse problems are considered. In this paper, key inequalities are established, which relate the approximation error with the modulus of continuity, as well as equivalence results for the K-functionals and the continuity module, and the role of Steklov eigenvalues in conditioning optimization problems discretized in weighted spaces. Numerical examples on standard problems in Operations Research support the results of this paper and show significant improvements in accuracy, conditioning, and efficiency over existing methods. This paper brings together abstract approximation theory and Operations Research optimization, which offers a firm foundation for the development of sophisticated computational methods in weighted function spaces.