Bounds for orthogonal polynomials for exponential weights
نویسنده
چکیده
Orthogonal polynomials pn(W ; x) for exponential weights W 2 = e−2Q on a nite or in nite interval I , have been intensively studied in recent years. We discuss e orts of the authors to extend and unify some of the theory; our deepest result is the bound |pn(W ; x)|W (x)|(x − a−n)(x − an)|6C; x∈ I with C independent of n and x. Here a±n are the Mhaskar–Rahmanov–Sa numbers for Q and Q must satisfy some smoothness conditions on I . c © 1998 Elsevier Science B.V. All rights reserved.
منابع مشابه
Solving singular integral equations by using orthogonal polynomials
In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...
متن کاملZeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights
We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form xγe−φ(x), with γ > 0, which include as particular cases the counterparts of the so-called Freud (i.e., when φ has a polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) weights. In ...
متن کاملUniform Asymptotics for Discrete Orthogonal Polynomials with Respect to Varying Exponential Weights on a Regular Infinite Lattice
Abstract. We consider the large-N asymptotics of a system of discrete orthogonal polynomials on an infinite regular lattice of mesh 1 N , with weight e , where V (x) is a real analytic function with sufficient growth at infinity. The proof is based on formulation of an interpolation problem for discrete orthogonal polynomials, which can be converted to a Riemann-Hilbert problem, and steepest de...
متن کاملAsymptotics associated with Exponential Weights
We announce some asymptotics for orthogonal and extremal polynomials associated with exponential weights W = exp ( Q). 1 Classes of Weights Let I be a nite or in nite interval and let Q : I ! [0;1) be convex. Let W := exp ( Q) and assume that all power moments Z I xW (x)dx; n = 0; 1; 2; 3; ::: are nite. Then we may de ne orthonormal polynomials pn(x) = pn(W ; x) = n(W )x + : : : ; n(W ) > 0;
متن کامل