On Best Error Bounds for Approximations by Algebraic Polynomials and Splines in the Spaces and

An analog of the well-known Jackson–Bernstein theory on best approximation by trigonometric polynomials is developed for approximation methods which use algebraic polynomials and piecewise polynomial functions on a finite interval. Approximation errors are measured in the norms of the Sobolev space and Besov space . These results can be useful in error analysis of the spectral, -, and - finite element methods for solving differential equations.