Sobolev-orthogonal systems of functions associated with an orthogonal system

For every system of functions {phi(k)(x)} which is orthonormal on (a, b) with weight rho(x) and every positive integer r we construct a new associated system of functions {phi(r,k)(x)}(k=0)(infinity) which is orthonormal with respect to a Sobolev-type inner product of the form < f, g > = Sigma(r-1)(nu=0)f((nu))(a)g((nu))(a) + integral(b)(a)f((r))(t)g((r))(t)rho(t) dt. We study the convergence of Fourier series in the systems {phi(r,k)(x)}(k=0)(infinity) In the important particular cases of such systems generated by the Haar functions and the Chebyshev polynomials T-n(x) = cos(n arccos x), we obtain explicit representations for the phi(r,k)(x) that can be used to study their asymptotic properties as k -> infinity and the approximation properties of Fourier sums in the system {phi(r,k)(x)}(k=0)(infinity). Special attention is paid to the study of approximation properties of Fourier series in systems of type {phi(r,k)(x)}(k=0)(infinity) generated by Haar functions and Chebyshev polynomials.