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 respect to the Sobolev-type inner product f, g = Σ r-1 ν=0 f (υ) (0) g (υ) (0)+ I ∞ 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 = 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.

Keywords

• Laguerre polynomials
• Laguerre functions
• inner product of Sobolev type
• Sobolev-orthonormal functions
• recurrence relations
• Fourier series
• asymptotic formula