Generalized Kronrod Patterson type imbedded quadratures

Error estimates for Gauss–Turán quadratures and their Kronrod extensions

We study the kernel Kn,s(z) of the remainder term Rn,s( f ) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L∞-error bounds of Gauss–Turán–Kronrod quadratures. Following Kronrod,...

On Chebyshev-Type Quadratures

According to a result of S. N. Bernstein, «-point Chebyshev quadrature formulas, with all nodes real, do not exist when n = 8 or n ä 10. Modifications of such quadrature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We...

On Generalized Gaussian Quadratures for Bandlimited Exponentials

We review the methods in [4] and [24] for constructing quadratures for bandlimited exponentials and introduce a new algorithm for the same purpose. As in [4], our approach also yields generalized Gaussian quadratures for exponentials integrated against a non-sign-definite weight function. In addition, we compute quadrature weights via l and l∞ minimization and compare the corresponding quadratu...

Simple universal bounds for Chebyshev-type quadratures

A Chebyshev-type quadrature for a probability measure σ is a distribution which is uniform on n points and has the same first k moments as σ. We give bounds for the smallest possible n required to achieve a certain degree k. In contrast to previous results of this type, our bounds use only simple properties of σ and are thus applicable in wide generality. In particular, it is shown that wheneve...

Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators