Theorems on Direct and Inverse Approximation by Algebraic Polynomials and Piecewise Polynomials in the Spaces \({{H}^{m}}(a,b)\) and \(B_{{2,q}}^{s}(a,b)\)

The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms of Sobolev and Besov spaces. We indicate the weighted Besov spaces, whose functions satisfy Jackson-type and Bernstein-type inequalities and, as a consequence, direct and inverse approximation theorems. In a number of cases, exact constants are indicated in the estimates.