Berry-ess Een-type Inequalities for Ultraspherical Expansions
نویسنده
چکیده
This paper contains several variants of Berry-Ess een-type inequalities for ultraspher-ical expansions of probability measures on 0; ]. Similar to the classical results on IR , proofs will be based in some cases on ultraspherical analogues of Fej er-kernels. The inequalities in this paper in particular lead to relations between the spherical-cap-distance of probability measures on unit spheres S d IR d+1 and the norms of associated L 2-convolution operators. Moreover, the inequalities will be used to derive the order of convergence for some central limit theorems on 0; ] and on S d ; the limit distributions there are analogues of Gaussian measures and the uniform distribution. 1 1. Introduction The classical Berry-Ess een-inequality relates the k:k 1 {distance of distribution func
منابع مشابه
The coefficients of differentiated expansions of double and triple Jacobi polynomials
Formulae expressing explicitly the coefficients of an expansion of double Jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultraspher...
متن کاملOn Estimates for the Weights in Gaussian Quadrature in the Ultraspherical Case
In this paper the Christoffel numbers av n for ultraspherical weight functions wk , wx(x) = (\ -x ) ~ ' , are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by dv „ the trigonometric representation of the Gaussian nodes, we obtain for À e [0, 1] the ineq...
متن کاملA Berry-Esseen Type Bound for the Kernel Density Estimator of Length-Biased Data
Length-biased data are widely seen in applications. They are mostly applicable in epidemiological studies or survival analysis in medical researches. Here we aim to propose a Berry-Esseen type bound for the kernel density estimator of this kind of data.The rate of normal convergence in the proposed Berry-Esseen type theorem is shown to be O(n^(-1/6) ) modulo logarithmic term as n tends to infin...
متن کاملTurán Type inequalities for (p, q)-Gamma function
have many applications in pure mathematics as in other branches of science. They are named by Karlin and Szegő in [8], Turán-type inequalities because the first of these type of inequalities was introduced by Turán in [18]. More precisely, he used some results of Szegő in [17] to prove the previous inequality for x ∈ (−1, 1), where fn is the Legendre polynomial of degree n. This classical resul...
متن کامل