We explore properties of Cauchy-Stieltjes families that have no counterpart in exponential families. We relate the variance function of the iterated Cauchy-Stieltjes family to the pseudo-variance function of the initial Cauchy-Stieltjes family. We also investigate when the domain of means can be extended beyond the ”natural domain”.

A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation (Dk+Yfcrn~JQ bmDm)f = 0 , where each bm is a complex number, D is the Dirac operator in R" , and f is defined on a domain in R" and takes values in a complex Clifford algebra. Some basic properties for the solutions to this equation, arising from the integral formula, are described, including an approximatio...

In this paper we present two different results in the theory of Cauchy-Stieltjes Kernel (CSK) families. We firstly provide construction free Sheffer systems with CSK associate a additive convolution semigroup probability measures to any and prove that is only one leads an orthogonal systems. also show orthogonality occurs if associated generates families quadratic variance function. Secondly, a...

We propose a discretized Tikhonov regularization for a Cauchy problem for an elliptic equation by a reproducing kernel Hilbert space. We prove the convergence of discretized regularized solutions to an exact solution. Our numerical results demonstrate that our method can stably reconstruct solutions to the Cauchy problems even in severe cases of geometric configurations.

The main purpose of this article is to present an approximation method of for singular integrodifferential equations with Cauchy kernel in the most general form under the mixed conditions in terms of the second kind Chebyshev polynomials. This method transforms mixed singular integro-differential equations with Cauchy kernel and the given conditions into matrix equation and using the zeroes of ...

In this paper we express the disentangling, or the formation of a function of several noncommuting operators using Cauchy’s Integral Formula in several complex variables. It is seen that the disentangling of a given function f can be expressed as a contour integral around the boundary of a polydisk where the standard Cauchy kernel is replaced by the disentangling of the Cauchy kernel expressed ...

Analytic functions in tubes in association with ultradistributional boundary values are analyzed. Conditions are stated on the analytic functions satisfying a certain norm growth which force the functions to be in the Hardy space H2 . Properties of the Cauchy kernel and Cauchy integral are obtained which extend results obtained previously by the author and collaborators.