Sobolev-orthogonal systems of functions associated with an orthogonal system

For every system of functions $\left\{{\phi }_k\right(x\left)\right\}$ which is orthonormal on $(a,b)$ with weight $\rho \left(x\right)$ and every positive integer $r$ we construct a new associated system of functions $\left\{{\phi }_{r,k}\right(x){\}}_{k=0}^{\infty }$ which is orthonormal with respect to a Sobolev-type inner product of the form We study the convergence of Fourier series in the systems $\left\{{\phi }_{r,k}\right(x){\}}_{k=0}^{\infty }$. In the important particular cases of such systems generated by the Haar functions and the Chebyshev polynomials $T_n\left(x\right)=\cos (n\arccos x)$, we obtain explicit representations for the ${\phi }_{r,k}\left(x\right)$ that can be used to study their asymptotic properties as $k\to \infty$ and the approximation properties of Fourier sums in the system $\left\{{\phi }_{r,k}\right(x){\}}_{k=0}^{\infty }$. Special attention is paid to the study of approximation properties of Fourier series in systems of type $\left\{{\phi }_{r,k}\right(x){\}}_{k=0}^{\infty }$ generated by Haar functions and Chebyshev polynomials. (paper)