On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

Abstract

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l_r,k^α(x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

\(\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt\)

, and generated by the classical orthogonal Laguerre polynomials L_k^α(x) (k = 0, 1,...). The polynomials l_r,k^α(x) are represented as expressions containing the Laguerre polynomials L_n^α−r (x). An explicit form of the polynomials l_r,k+r^α(x) is established as an expansion in the powers x^r+l, l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l_r,k^α(x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.