Step-by-step solving of ordered interpolation problem for stieltjes functions

We investigate a step-by-step solving of ordered generalized interpolation problems for Stieltjes matrix functions and obtain a multiplicative representation for the sequence of resolvent matrices. Thematrix factors inmultiplicative representations of the resolventmatrices are expressed through the Schur–Stieltjes parameters, for which we obtain explicit formulas and give an algorithm of step-by-step solving of Stieltjes type interpolation problems. As examples, we consider step-by-step solutions of the Stieltjes matrix moment problem and the problems by Nevanlinna–Pick and Caratheodory.     

The Schur algorithm for an indefinite Stieltjes moment problem

The nondegenerate truncated indefinite Stieltjes moment problem in the class of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step‐by‐step algorithm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recently in [11]. Explicit formula for the resolvent matrix in terms of generalized Stieltjes polynomials is found.     

Stieltjes moment problem and fractional moments

Stieltjes moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, a determinate Stieltjes moment problem, whose corresponding Hamburger moment problem is determinate too, is investigated in the setup of Maximum Entropy. Condition number in entropy calculation is provided endowing both Stieltjes moment problem existence conditions and Hamburger moment problem determinacy conditions by a geometric meaning. Then the resorting to fractional moments is considered; numerical aspects are investigated and a stable algorithm for calculating fractional moments from integer moments is proposed.     

An algorithm for determining if the inverse of a strictly diagonally dominant Stieltjes matrix is strictly ultrametric

Summary It was recently shown that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. On the other hand, as it is well-known that the inverse of a strictly diagonally dominant Stieltjes matrix is a real symmetric matrix with nonnegative entries, it is natural to ask, conversely, if every strictly diagonally dominant Stieltjes matrix has a strictly ultrametric inverse. Examples show, however, that the converse is not true in general, i.e., there are strictly diagonally dominant Stieltjes matrices in ^n×n (for everyn3) whose inverses are not strictly ultrametric matrices. Then, the question naturally arises if one can determine which strictly diagonally dominant Stieltjes matrices, in ^n×n (n3), have inverses which are strictly ultrametric. Here, we develop an algorithm, based on graph theory, which determines if a given strictly diagonally dominant Stieltjes matrixA has a strictly ultrametric inverse, where the algorithm is applied toA and requires no computation of inverse. Moreover, if this given strictly diagonally dominant Stieltjes matrix has a strictly ultrametric inverse, our algorithm uniquely determines this inverse as a special sum of rank-one matrices.Research supported by the National Science FoundationResearch supported by the Deutsche Forschungsgemeinschaft     

Structure of Stieltjes classes of moment-equivalent probability laws

A Stieltjes class is a one-parameter family of moment-equivalent distribution functions constructed by modulation of a given indeterminate distribution function F, called the center of the class. Members of a Stieltjes class are mutually absolutely continuous, and conversely, any pair of moment-equivalent and mutually absolutely continuous distribution functions can be joined by a Stieltjes class. The center of a Stieltjes class is an equally weighted mixture of its extreme members, and this places restrictions on which distributions can belong to a Stieltjes class with a given center. The lognormal law provides interesting illustrations of the general ideas. In particular, it is possible for two moment equivalent infinitely divisible distributions to be joined by a Stieltjes class, and random scaling can be used to construct new Stieltjes classes from a given Stieltjes class.     

Vector Stieltjes Continued Fraction and Vector QD Algorithm

The definition, in previous studies, of vector Stieltjes continued fractions in connexion with spectral properties of band operators with intermediate zero diagonals, left unsolved the question of a direct definition of their coefficients in terms of the original data, a vector of Stieltjes series. A new version of the vector QD algorithm allows to extend to the vector case the result which was known for one scalar function. Beside this connexion, it solves the inverse Miura transform and gives interesting identities between general band matrix and sparse band matrix. It gives also some effective computations of coefficients linked to vector orthogonality.     

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.     

A generalized Stieltjes criterion for the complete indeterminacy of interpolation problems

The main result of this paper is a generalized Stieltjes criterion for the complete indeterminacy of interpolation problems in the Stieltjes class. This criterion is a generalization to limit interpolation problems of the classical Stieltjes criterion for the indeterminacy of moment problems. It is stated in terms of the Stieltjes parameters M _j and N _j . We obtain explicit formulas for the Stieltjes parameters. General constructions are illustrated by examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class.     

On the relation between measures defining the Stieltjes and the inverted Stieltjes functions

A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function.     

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.     

A simple algorithm for expanding a power series as a continued fraction

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).     

A note on a Mar?enko-Pastur type theorem for time series

In this note we develop an extension of the Mar?enko-Pastur theorem to time series model with temporal correlations. The limiting spectral distribution (LSD) of the sample covariance matrix is characterised by an explicit equation for its Stieltjes transform depending on the spectral density of the time series. A numerical algorithm is then given to compute the density functions of these LSD’s.     

Linear Stieltjes equation with generalized riemann integral and existence of regulated solutions

1. IntroductiollIn the last two decades the memann generalized integral, having values in B-spaces,has been increasingly studied.The development in this area mainly concerns the HenstoCk-Kurzweil (see e.g. I1, 2]),and the Dushnik and the Young integrals (see e.g. I3]). Recently there appeared in the1iterature many proper app1ications in this field ("proper" here being considered in thesense that the results are not disguises of an essentially finite dimensional frame) (see e.g.[4, 5]).In t…     

Moment problem of Hamburger,hierarchies of integrable systems,and the positivity of tau-functions

A moment problem of Hamburger is studied to find a parametric Stieltjes measure from given moments. It is shown that if a deformation, or a dynamics, of moments is governed by a hierarchy of a Kac-van Moerbeke system, then the Stieltjes measure can be constructed explicitly by integrating a hierarchy of Moser's nonlinear dynamical system. The positivity of tau-functions is related to the existence of the Stieltjes measure at a deep level.     

Addition in a class of nonlinear Stieltjes integrators

A product integral formula is established for the generation of an evolution system by the sum of two Stieltjes integrators. The class of Stieltjes integrators considered is the class recently introduced and studied by J.V. Herod.     

Indeterminacy Criteria for the Stieltjes Matrix Moment Problem

In this paper, we obtain criteria for the indeterminacy of the Stieltjes matrix moment problem. We obtain explicit formulas for Stieltjes parameters and study the multiplicative structure of the resolvent matrix. In the indeterminate case, we study the analytic properties of the resolvent matrix of the moment problem. We describe the set of all matrix functions associated with the indeterminate Stieltjes moment problem in terms of linear fractional transformations over Stieltjes pairs.     

General existence principles for Stieltjes differential equations with applications to mathematical biology

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.     

On functions which preserve the class of Stieltjes matrices

A positive definite symmetric matrix is called a Stieltjes matrix provided that all its off diagonal elements are nonpositive. We characterize functions which preserve the class of Stieltjes matrices.     

The approximation of Laplace–Stieltjes transformations with finite order on the left half plane

In this paper, we study the error in approximating the analytic function defined by a Laplace–Stieltjes transformation of finite order, which converges on the left half plane, and obtain the relation theorems between the error, the coefficients, and the proximate order of the Laplace–Stieltjes transformation.     

On the Galois groups of Legendre polynomials

Ever since Legendre introduced the polynomials that bear his name in 1785, they have played an important role in analysis, physics and number theory, yet their algebraic properties are not well-understood. Stieltjes conjectured in 1890 how they factor over the rational numbers. In this paper, assuming Stieltjes’ conjecture, we formulate a conjecture about the Galois groups of Legendre polynomials, to the effect that they are “as large as possible,” and give theoretical and computational evidence for it.