Approximation in certain intermediate spaces

Abstract

A theorem of Bojanic gives a precise estimate on the rate of convergence of the Fourier series of a function of boundend variation. While the method of K-functionals is not directly applicable to obtain similar estimates for functions in classes intermediate to BV[−1, 1] and C[−1, 1]. we obtain such an estimate in the case of a general class of operators. The result is given in terms of an expression, which for continuous functions, is equivalent to the K-functional. As particular cases, we study the expansions in certain (general) orthogonal polynomials, Lagrange interpolation at the zeros of (general) orthogonal polynomials, and Hermite-Fejér interpolation at the zeros of generalized Jacobi polynomials. When applicable, our result (essentially) includes the previously known results, while many corollaries are new.