Asymptotics for Entropy Integrals associated with Exponential Weights
نویسنده
چکیده
We establish a rst order asymptotic for the entropy integrals Z I pn log pn W 2 and Z I pn log (pnW ) 2 W 2 where fpngn=0 are the orthonormal polynomials associated with the exponential weight W . 1 The Result Let I = (c; d) be a real interval, where 1 c < 0 < d 1; 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) = nx n + :::; n > 0;
منابع مشابه
Asymptotics 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;
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