Baskakov operators and Jacobi weights: pointwise estimates

In this paper we present direct results (upper estimates) for Baskakov operators acting in spaces related with Jacobi-type weights. Our results include and extend some known facts related with this problem. The approach is based in the use of a new pointwise K-functional.

Linear differential equations for the resolvents of the classical matrix ensembles

The spectral density for random matrix [Formula: see text] ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of [Formula: see text], which for even [Formula: see text] is a polynomial of degree [Formula: see text]. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution of a linear differential equation of degree [Formula: see text]. This equation, and its companion for the resolvent, are given explicitly for [Formula: see text] and [Formula: see text] for all three classical cases, and also for [Formula: see text] in the Gaussian case. Known dualities for the spectral moments relating [Formula: see text] to [Formula: see text] then imply corresponding differential equations in the case [Formula: see text], and for the Gaussian ensemble, the case [Formula: see text]. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their [Formula: see text] expansions, along with first-order differential equations for the coefficients of the [Formula: see text] expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.

On the Convergence Rate of Clenshaw–Curtis Quadrature for Jacobi Weight Applied to Functions with Algebraic Endpoint Singularities

Applying the aliasing asymptotics on the coefficients of the Chebyshev expansions, the convergence rate of Clenshaw–Curtis quadrature for Jacobi weights is presented for functions with algebraic endpoint singularities. Based upon a new constructed symmetric Jacobi weight, the optimal error bound is derived for this kind of function. In particular, in this case, the Clenshaw–Curtis quadrature for a new constructed Jacobi weight is exponentially convergent. Numerical examples illustrate the theoretical results.