Orthogonal Polynomials for Exponential Weights by Eli Levin, Doron S. Lubinsky (auth.)

By Eli Levin, Doron S. Lubinsky (auth.)

The research of orthogonal polynomials linked to common weights was once a huge subject in classical research within the 20th century, and certainly will keep growing in value within the future.
In this monograph, the authors examine orthogonal polynomials for exponential weights outlined on a finite or limitless period. The period may still comprise zero, yet needn't be symmetric approximately zero; likewise the load needn't be even. The authors identify bounds and asymptotics for orthonormal and extremal polynomials, and their linked Christoffel features. They deduce bounds on zeros of extremal and orthogonal polynomials, and in addition identify Markov- Bernstein and Nikolskii inequalities.
The authors have collaborated actively because 1982 on quite a few themes, and feature released many joint papers, in addition to a Memoir of the yankee Mathematical Society. The latter offers with a different case of the weights handled during this ebook. in lots of methods, this booklet is the end result of 18 years of joint paintings on orthogonal polynomials, drawing thought from the works of many researchers within the very lively box of orthogonal polynomials.

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X2n < Xln < d. 19 Let W E F (lip ~+). 5), as well as further results of this type, including estimates of the Lp norms of orthonormal polynomials. 3). 5 Asymptotics of Extremal and Orthonormal Polynomials The Lp extremal error associated with monic polynomials of degree n is En,p(W):= inf PE'Pn - 1 \I [xn - P(x)] W(X)\\Lp(I). In the special case P = 2, it is well known that E n ,2(W) is the reciprocal of the leading coefficient of Pn (W2 ,. ). 20 Let W E F (dini) and 1 :::; P :::; K,p:={ 00. Let [J1iT(~)/r(~+I)]l/P, 1, Then as n -+ 00, we have (6 En,p(W) = 2K,p; )n+l/p exp (li -1[' an a_ n p

Lq). 9)). 10). Lq) belongs to the interior of I, we deduce that it has the required form. 201-2021 so we only sketch it. For 0, (3 E I, 0 < (3, let F(o,(3) = 11(3 (3 - 0 log-- - 4 11' (3 -- 0 - -1 Io g 4 11' 11 Q -1 vi dx (x - 0)((3 - x) q(x)---;=j=~~=7 0 q ((3 -+2 (3 --0) +t 2 dt v'f=t2 . 2) that maxF(o,(3) = F(a_q,aq). 11). e. on 1 and is in Loo(~q). 11) exist. Finally, as q is convex, the function (3+0 (3-0) (0,(3) -+ -q ( -2- + -2- t is concave, for any fixed t E [-1,11. Since the function 10g((3 - 0)/4 is strictly concave, we see from the second expression for F(o, (3) above, that F is strictly concave.

118) and its cosine cousin, W~(O) := W~(cosO), 0 E [-71",71"). 21 Let W E :F (dini). (/) Let 1 < P < We have 00. 120) (II) Let 1 < P ::; n --+ 00, 00. Uniformly for z in closed subsets of «::\[-1, 1), as «5~/pPn,p(W, Lh- 1](z)) / = 2 1/ p - { ¢(z)n D- 2 (w~; ¢tZ)) (1 _ ¢(z)-2) -l/p } 1 --(1 + 0(1)). 121) "'p By specializing to P = 2, one obtains asymptotics in the plane and mean asymptotics for the orthonormal polynomials Pn(x) = Pn(W 2,x) = 'Yn (W2) xn + ... 1). 22 Let W E :F (dini). Then as n --+ 'Yn (W2) = 1 «5 fiC ( ; ) V 271" -n- 2 1 exp 00, ( -1 1 an 71" a- n s Q( ) J(s - a_ n ) (an - s) ds ) (1+0(1)).

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