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

We consider the polynomials T-r,T-n(x) (n = 0, 1,...) generated by Chebyshev polynomials T-n(x) and forming a Sobolev orthonormal system with respect to the inner productf, g = r- 1 .= 0 f(.)(- 1) g(.)(- 1) + 1 - 1 f(r)(x) g(r)(x) mu(x) dx,, where (x) = 2(-1)(1 - x(2))(-1/2). It is shown that the Fourier sums in the polynomials T-r,T-n(x) (n = 0, 1,...) give a convenient and efficient tool for approximately solving the Cauchy problem for ordinary differential equations.