Sobolev Orthogonal Polynomials Associated with Chebyshev Polynomials of the First Kind and the Cauchy Problem for Ordinary Differential Equations

Abstract

We consider the polynomials T_r,n(x) (n = 0, 1,…) generated by Chebyshev polynomials T_n(x) and forming a Sobolev orthonormal system with respect to the inner product

\(\langle f,g\rangle = \sum\limits_{\nu = 0}^{r - 1} {{f^{(\nu)}}} ( - 1){g^{(\nu)}}(-1) + \int\limits_{-1}^1 {{f^{(r)}}} (x){g^{(r)}}(x)\mu (x)dx,\)

, where μ(x) = 2π⁻¹(1 − x²)^−1/2. It is shown that the Fourier sums in the polynomials T_r,n(x) (n = 0, 1,…) give a convenient and efficient tool for approximately solving the Cauchy problem for ordinary differential equations.