Sobolev-orthogonal systems of functions associated with an orthogonal system

I. I. Sharapudinov

Izvestiya. Mathematics , Tome 82 (2018) no. 1, pp. 212-244

Résumé

For every system of functions $\{\varphi_k(x)\}$ which is orthonormal on $(a,b)$ with weight $\rho(x)$ and every positive integer $r$ we construct a new associated system of functions $\{\varphi_{r,k}(x)\}_{k=0}^\infty$ which is orthonormal with respect to a Sobolev-type inner product of the form $$ \langle f,g \rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+ \int_{a}^{b} f^{(r)}(t)g^{(r)}(t)\rho(t) \,dt. $$ We study the convergence of Fourier series in the systems $\{\varphi_{r,k}(x)\}_{k=0}^\infty$. 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 $\varphi_{r,k}(x)$ that can be used to study their asymptotic properties as $k\to\infty$ and the approximation properties of Fourier sums in the system $\{\varphi_{r,k}(x)\}_{k=0}^\infty$. Special attention is paid to the study of approximation properties of Fourier series in systems of type $\{\varphi_{r,k}(x)\}_{k=0}^\infty$ generated by Haar functions and Chebyshev polynomials.

Détail

Keywords: Sobolev-orthogonal systems of functions associated with Haar functions; Sobolev-orthogonal systems of functions associated with Chebyshev polynomials; convergence of Fourier series of Sobolev-orthogonal functions; approximation properties of partial sums of Fourier series of Sobolev-orthogonal functions; convergence of Fourier series of Sobolev-orthogonal polynomials associated with Chebyshev polynomials.