Asymptotics of Sobolev Orthogonal Polynomials for Coherent Pairs of Measures

Sobolev orthogonal polynomials: Balance and asymptotics

“…We denote by P n,µ 0 , P n,µ 1 and S n,λ the corresponding monic polynomials orthogonal with respect to µ 0 , µ 1 and ·, · λ , respectively. Let µ 0 and µ 1 be measures compactly supported on R. Whether (µ 0 , µ 1 ) is a coherent pair, which means that there exist nonzero constants σ n such that the corresponding monic polynomials satisfy for each n, P n,µ 1 = P n+1,µ 0 n + 1 + σ n P n,µ 0 n or, if µ 0 and µ 1 fulfill much milder conditions, i.e., they belong to the well-known Szegő class, then it has been established (see [9] and [8]) that the ratio asymptotics Nevertheless, a closer look at the inner product (1) explains the "dominance" of the measure µ 1 in the asymptotics: the derivative makes the leading coefficient of the polynomials in the second integral of (1) be multiplied by the degree of the polynomial. Thus, if we want both measures to have an impact on the behavior of the polynomials for n → ∞, it seems natural to "balance" the inner product, that is, to compensate both integrals by introducing a varying parameter λ n .…”

Asymptotics for Sobolev Orthogonal Polynomials for Exponential Weights

“…(3) Sobolev orthogonal polynomials have found application in a number of contexts. See, for instance, the survey papers [19], [23], as well as [16]. In particular in [13], a study of Fourier series of Sobolev orthogonal polynomials was initiated for smooth functions.…”

“…If we do not assume it, we can still prove a version of (23) in which Q is replaced by a function that grows more slowly as we approach the endpoints of I .…”

“…Consequently, the analytical properties of the Sobolev orthogonal polynomials could be determined from those of classical orthogonal polynomials. This investigation was continued in [23] for the bounded (Jacobi) case and in [20] for the unbounded Laguerre case.…”

Companion Linear Functionals and Sobolev Inner Products: A Case Study

“…The relation (1.6) has been used as a basic tool for analysis of asymptotic properties of the polynomials {Q n (x; λ)} (see for instance [8,12,13,15]). …”