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In this paper, we establish some inequalities which improve some Ostrowski type inequalities. Applications for Euler’s Beta mapping and special means are also given.

Asymptotic expressions for remainder terms of the mid-point, trapezoid and Simpson’s rules are given. Corresponding formulas with finite sums are also given.

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code - a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values - and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups., Comment: 25 pages, no figures

We derive new formulas for harmonic, diffraction, elastic, and hydrodynamic potentials acting on anisotropic Gaussians and approximate wavelets. These formulas can be used to construct accurate cubature formulas for these potentials.

Some inequalities and approximations for the finite Hilbert transform by the use of Taylor's formula with the integral remainder are given.

In this article, with the help of concept of the harmonic sequence of polynomials, the well known Hermite-Hadamard’s inequality for convex functions is generalied to the cases with bounded derivatives of n-th order, including the so-called n-convex functions, from which Hermite-Hadamard’s inequality is extended and refined.

We investigate approximation methods and the stability of a class of integral equations on the line for Freud exponential weights.

We study the convergence of quadrature formulas for integrals over the positive real line with an arbitrary distribution function. The nodes of the quadrature formulas are the zeros of orthogonal Laurent polynomials with respect to the distribution function and with respect to a certain nesting. This ensures a maximal domain of validity and the quadratures are therefore called Gauss-type formulas. The class of functions for which convergence holds is characterized in terms of the moments of the distribution function. Moreover, error estimates are given when f satisfies certain continuity conditions. Finally, these results are applied to the family of distributions dφ(x) = x^α exp{-(x^(γ_1) + x^(-γ_2))) dx, γ_1,γ_2 greater than or equal to 1/2, α is an element of R. editors: Bonan-Hamada, Cathy and Gustafson, Phil ispartof: The Rocky Mountain journal of mathematics vol:33 issue:2 pages:585-608 ispartof: location:Mesa State College, Grand Junction, CO status: published

An optimal 3-point quadrature formula of closed type is derived. Various error inequalities are established. Applications in numerical integration are also given.

An error analysis for some Newton-Cotes quadrature formulae is presented. Peano-like error bounds are obtained. They are generally, but not always, better than the usual Peano bounds.

We study linear estimators for the weighted integral of a stochastic process. The process may only be observed on a finite sampling design. The error is defined in a mean square sense, and the process is assumed to satisfy Sacks-Ylvisaker regularity conditions of order $r \epsilon \mathbb{N}_0$. We show that sampling at the quantiles of a particular density already yields asymptotically optimal estimators. Hereby we extend the results of Sacks and Ylvisaker for regularity $r = 0$ or 1, and we confirm a conjecture by Eubank, Smith and Smith.

1. Introduction. We study multivariate integration and L2-approximation for random fields Y that are defined on the d-dimensional unit cube D = [0, 1]d and that have mean zero and known covariance kernel K. We assume that K is at least continuous, and hence we may assume that Y is a measurable random field whose realizations are in L2(D) with probability 1. For integration we want to estimate the integral fD Y(t) dt, whereas for L2approximation we want to estimate the values Y(t) for all t, and we study the distance of the estimate and the realization of the field in the space L2(D). For both problems we mainly consider linear estimators that use n observations of the random field. These estimators are of the form n n

Consider predicting an integral of a stochastic process based on $n$ observations of the stochastic process. Among all linear predictors, an optimal quadrature rule picks the $n$ observation locations and the weights assigned to them to minimize the mean squared error of the prediction. While optimal quadrature rules are usually unattainable, it is possible to find rules that have good asymptotic properties as $n \rightarrow \infty$. Previous work has considered processes whose local behavior is like $m$-fold integrated Brownian motion for $m$ a nonnegative integer. This paper obtains some asymptotic properties for quadrature rules based on median sampling for processes whose local behavior is not like $m$-fold integrated Brownian motion for any $m$.

The paper discusses new cubature formulas for classical integral operators of mathematical physics based on the "approximate approximation" of the density with Gaussian and related functions. We derive formulas for the cubature of harmonic, elastic and diffraction potentials approximating with high order in some range relevant for numerical computations. We prove error estimates and provide numerical results for the Newton potential.

We study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical Wiener sheet measure. To derive the average case complexity one needs to obtain optimal sample points. We prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years. This relation enables us to show that optimal sample points can be derived from Hammersley points. Extending the result of Roth and using the recent result of Wasilkowski, we conclude that the average case complexity is θ(e -1 (lne -1 ) (d-1)/2 )

In this paper we investigate the relationship between total positivity and reproducing kernels. We extend the notion of total positivity to domains in the complex plane. In doing so, we also give a geometrical interpretation to certain Wronskians of reproducing kernels. These geometrical quantities are connected to Gaussian curvatures of Kahler metrics induced by these kernels. For simply-connected domains these curvatures are negative constants, thereby showing that the kernels are totally positive and moreover yielding an efficient method for computing the relevant determinants. In general, the reproducing kernels of multiplyconnected domains are not totally positive. The motivation for this work stems from the work of Karlin [7] which deals with "optimal" quadrature formulas. Let H be a Hubert space of functions analytic in a plane domain D and possessing a reproducing kernel K(z, t), z,teD. Let LeH*, where iϊ* is the dual of H. A subset & of H* is specified and a member Q e & is called a quadrature formula. To each Q e & is associated a remainder functional RQ = L — Q. An optimal quadrature formula, if it exists, is any member Q* e ^ satisfying

Some new Iyengar type inequalities for an integral are obtained by using the generalised Taylor formula with integral remainder.