Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss-Radau (m = 1) and Gauss-Lobatto (m = 2) quadrature formulas that approximate F{f}. These are quadrature formulas with n positive weights and with the n−m remaining nodes real a...

We show how to combine incidence matrices, which admit Hermite-Birkhoff quadrature formulas of Gaussian type for any positive measure, in such a way that the resulting matrix also admits Gaussian type quadratures for any positive measure. Moreover, the uniqueness property and the extremal property of the formulas corresponding to the submatrices are transferred to the formula admitted by the co...

In the framework of Richards–Wolf formalism, spin–orbit conversion upon tight focusing an optical vortex with circular polarization is studied. We obtain exact formulas which show what part total (averaged over beam cross-section) longitudinal spin angular momentum transferred to orbital in focus. It shown that maximum can be transformed into equal half power, and this reached at numerical aper...

Abstract. A series of papers have treated the frequency analysis problem by studying the zeros of orthogonal polynomials on the unit circle with respect to measures determined by observations of the signal. In the recent paper [3], a different approach was used, where properties of Szegő quadrature formulas associated with the zeros of paraorthogonal polynomials with respect to the same measure...

A generalization of the pre-Grüss inequality is presented. It is applied to estimations of remainders of some quadrature formulas.

In this paper, we will derive several error inequalities for a quadrature formula with a parameter, which will not only provide some generalizations of the known results, but also give some other interesting quadrature formulae as special cases. Furthermore, sharp upper and lower error bounds for the double error inequalities are obtained. Applications in numerical integration are also given. ©...

We study the construction of a quadrature rule which allows the simultaneous integration of a given function with respect to different weights. This construction is built on the basis of simultaneous Padé approximation of a Nikishin system of functions. The properties of these approximants are used in the proof of convergence of the quadratures and positivity of the corresponding quadrature coe...

An evaluation of the derivative of spherical Bessel functions of order n + 1 2 at its zeros is obtained. Consequently, an explicit quadrature formula for entire functions of exponential type is given.

The option-pricing problem is always an important part in modern finance. Assuming that the stock diffusion is a constant, some literature has introduced many stock models and given corresponding option pricing formulas within the framework of the uncertainty theory. In this paper, we propose a new stock model with uncertain stock diffusion for uncertain markets. Some option pricing formulas on...