Asymptotics of Sobolev orthogonal polynomials for Hermite (1,1)-coherent pairs

作者：Duenas Ruiz Herbert; Marcellan Espanol Francisco; Molano Molano Alejandro

来源：Journal of Mathematical Analysis and Applications, 2018, 467(1): 601-621.

DOI：10.1016/j.jmaa.2018.07.030

• Hermite (1,1)-coherent pairs
• Sobolev polynomials
• Asymptotic properties

In this paper we will discuss asymptotic properties of monic polynomials {S-n(lambda)(x)}n >= 0 orthogonal with respect to the Sobolev inner product
< p, q >(S) = integral(R) p(x)q(x)d mu(0) + lambda integral(R) p'(x)q'(x)d mu(1),
with lambda > 0, d mu(0) = e(-x2) dx, d(mu 1), d(mu 1) = x(2)+a/x(2)+b e(-x2) dx, a, b is an element of R+ and a not equal b. It is well known that (mu 0, mu 1) is a pair of symmetric (1, 1)-coherent measures. This means that there exist sequences {a(n)}(n is an element of N), {b(n)}(n is an element of N), a(n) not equal b(n) for every n is an element of N, such that the algebraic relation
H-n(x) + b(n-2)H(n-2)(x) = Q(n)(x) + a(n-2)Q(n-2)(x), n >= 2,
is satisfied, where {Q(n)(x)}(n >= 0) is the sequence of monic orthogonal polynomials associated with mu(1) and {H-n(x)(n >= 0) is the sequence of monic Hermite polynomials. We will study the relative asymptotics for Sobolev scaled polynomials and we will obtain Mehler Heine type formulas, among others.

• 出版日期2018-11-1