ISSN:
1432-0940
Keywords:
Primary 41A25
;
Primary 42C05
;
Exponential weights
;
Freud's conjecture
;
Orthogonal polynomials
;
Recurrence relation coefficients
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW 2(x) are finite. Let {p n (W 2;x)} 0 ∞ denote the sequence of orthonormal polynomials with respect to the weightW 2(x), and let {A n } 1 ∞ and {B n } 1 ∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = A_{n + 1} p_{n + 1} (W^2 ,x) + B_n p_n (W^2 ,x) + A_n p_{n - 1} (W^2 ,x).$$ . WhenW(x) =w(x) exp(-Q(x)), xε(-∞,∞), wherew(x) is a “generalized Jacobi factor,” andQ(x) satisfies various restrictions, we show that $$\mathop {\lim }\limits_{n \to \infty } {{A_n } \mathord{\left/ {\vphantom {{A_n } {a_n }}} \right. \kern-\nulldelimiterspace} {a_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{B_n } \mathord{\left/ {\vphantom {{B_n } {a_n }}} \right. \kern-\nulldelimiterspace} {a_n }} = 0,$$ where, forn large enough,a n is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {a_n xQ'(a_n x)(1 - x^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} dx.}$$ In the special case, Q(x) = ¦x¦α, a 〉 0, this proves a conjecture due to G. Freud. We also consider various noneven weights, and establish certain infinite-finite range inequalities for weighted polynomials inL p(R).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02075448
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