Sobolev-Orthonormal System of Functions Generated by the System of Laguerre Functions

Abstract

We consider the system of functions λ α r,n (x) (r ∈ N, n = 0, 1, 2,. . .), orthonormal with respect to the Sobolev-type inner product f, g = r−1 ν=0 f (ν) (0)g (ν) (0)+ ∞ 0 f (r) (x)g (r) (x)dx and generated by the orthonormal Laguerre functions. The Fourier series in the system {λ α r,n (x)} ∞ k=0 is shown to uniformly converge to the function f ∈ W r L p for 4 3 < p < 4, α 0, x ∈ [0, A], 0 A < ∞. Recurrence relations are obtained for the system of functions λ α r,n (x). Moreover, we study the asymptotic properties of the functions λ α 1,n (x) as n → ∞ for 0 x ω, where ω is a fixed positive real number.