Stieltjes Class Research Articles

Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

Probability densities that are not uniquely determined by their moments are said to be “moment-indeterminate,” or “M-indeterminate.” Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.

Representations of closed quadratic forms associated with Stieltjes and inverse Stieltjes holomorphic families of linear relations

In this paper holomorphic families of linear relations that belong to the Stieltjes or inverse Stieltjes class are studied. It is shown that in their domain of holomorphy $\mathbb{C}\setminus\mathbb{R}_+$ the values of Stieltjes and inverse Stieltjes families are, up to a rotation, maximal sectorial. This leads to a study of the associated closed sesquilinear forms and their representations. In particular, it is shown that the Stieltjes and inverse Stieltjes holomorphic families of linear relations are of type (B) in the sense of Kato. These results are proved by using linear fractional transforms which connect these families to holomorphic functions that belong to a combined Nevanlinna-Schur class and a key tool then relies on a specific structure of contractive operators. <br><br> Розглядаються голоморфні сім’ї лінійних відношень, які належать до класу Стілтьєса та оберненого класу Стілтьєса. Показано, що в їхній області голоморфності $\mathbb{C}\setminus\mathbb{R}_+$ значення цих сімей є, з точністю до обертання, максимальними секторіальними. Із цим пов’язане дослідження відповідних замкнених півторалінійних форм та їхніх представлень. Зокрема, показано, що стілтьєсівські та обернені стілтьєсівські голоморфні сім’ї лінійних відношень належать до типу (В) у сенсі Като. Доведення базується на використанні дробово-лінійних перетворень, які переводять розглядувані сім’ї в голоморфні функції класу Неванлінни-Шура, псля чого використовується спеціальні структури операторів стиску.

Regularization of the indefinite Stieltjes moment problem

A not regular indefinite Stieltjes moment problem is studied in the generalized Stieltjes class Nκk. The new class of the sequence s={si}i=0ℓ is defined, which is called α-regular class. Regularization of the indefinite Stieltjes moment problem is obtained. The set of solutions of this problem is found in terms of the generalized Stieltjes polynomials and S-fractions with the shift α.

Resolvent Matrix of the Truncated Nevanlinna-Pick Matrix Interpolation Problem via Orthogonal Rational Functions in the Stieltjes Class

Resolvent Matrix of the Truncated Nevanlinna-Pick Matrix Interpolation Problem via Orthogonal Rational Functions in the Stieltjes Class

Резольвентная матрица усеченной матричной интерполяционной проблемы Неванлинны-Пика и ортогональные рациональные функции класса Стилтьеса

A new representation of the resolvent matrix and extremal solutions of the truncated Nevanlinna-Pick matrix interpolation problem in the Stieltjes class functions via orthogonal matrix rational functions on [0, ∞) is given.

An operator approach to the indefinite Stieltjes moment problem

A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class N κ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel $$ {\mathbf{N}}_{\omega }(z)\coloneq \frac{f(z)-\overline{f\left(\omega \right)}}{z-\overline{\omega}} $$ has κ negative squares on ℂ+. The generalized Stieltjes class $$ {\mathbf{N}}_{\kappa}^k\left(\kappa, k\in {\mathrm{\mathbb{Z}}}_{+}\right) $$ is defined as the set of functions f ϵ N κ such that z f ϵ N k . The full indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ consists in the following: Given κ, k ϵ ℤ+, and a sequence $$ \mathbf{s}={\left\{{s}_i\right\}}_{i=0}^{\infty } $$ of real numbers, to describe the set of functions $$ f\in {\mathbf{N}}_{\kappa}^k $$ , which satisfy the asymptotic expansion $$ f(z)=-\frac{s_0}{z}-\cdots -\frac{s_2n}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right)\kern1em \left(z=-y\in {\mathrm{\mathbb{R}}}_{-},y\uparrow \infty \right) $$ for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A [0;N] generated by $$ {\mathfrak{J}}_{\left[0;N\right]} $$ . The u-resolvent matrices of the operator A [0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ to be solvable and indeterminate are found. Explicit formulae for Pade approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.

Applications of the Stieltjes and Laplace transform representations of the hypergeometric functions

ABSTRACTIn our previous work, we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences of these properties. In particular, we find new integral representations of the generalized hypergeometric functions, evaluate a number of integrals of their products, compute the jump and the average value of the generalized hypergeometric function over the branch cut , and establish new inequalities for this function in the half-plane . Furthermore, we discuss integral representations of absolutely monotonic functions and present a curious formula for a finite sum of products of gamma ratios as an integral of Meijer's G function.

On a simultaneous approach to the even and odd truncated matricial Stieltjes moment problem II: An α-Schur–Stieltjes-type algorithm for sequences of holomorphic matrix-valued functions

The main goal of this paper is to achieve a simultaneous treatment of the even and odd truncated matricial Stieltjes moment problems in the most general case. These results are generalizations of results of Chen and Hu [5,19] which considered the particular case α=0. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version was worked out in a former paper of the authors. It is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and investigation of the function-theoretic version of our Schur-type algorithm is a central theme of this paper. This algorithm will be applied to relevant subclasses of holomorphic matrix-valued functions of the Stieltjes class. Using recent results on the holomorphicity of the Moore–Penrose inverse of matrix-valued Stieltjes functions, we obtain a complete description of the solution set of the moment problem under consideration in the most general situation.

Criterion for complete indeterminacy of limiting interpolation problem of Stieltjes type in terms of orthonormal matrix-functions

The main goal of the paper is to study the ordered sequences of generalized interpolation problems and the limiting interpolation problem in the Stieltjes class by means of the orthonormal matrix-functions. We obtain explicit formulas for the orthonormal matrix-functions. The main result of the paper is a criterion for the complete indeterminacy of limiting interpolation problem in the Stieltjes class. General constructions are illustrated by examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class. DOI: 10.3103/S1066369X15040015

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 Mj and Nj. 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.

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.

Indeterminacy of interpolation problems in the Stieltjes class

The concept of ordered families of interpolation problems in the Stieltjes class is introduced. Ordered families are used for the introduction of the concept of limiting interpolation problem in the same class. The limiting interpolation problem is proved to be soluble. A criterion for the complete indeterminacy of a limiting interpolation problem in the Stieltjes class is obtained. All solutions in the completely indeterminate case are described in terms of linear fractional transformations. General constructions are illustrated by the examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class.

Stieltjes classes for M-indeterminate powers of inverse Gaussian distributions

The aim of this paper is to exhibit an infinite family (Stieltjes class) of distributions all of which have the same moments as some powers of the inverse Gaussian distribution. For some particular cases of Stieltjes classes we have found the value of the index of dissimilarity.

New Stieltjes classes involving generalized gamma distributions

Let F be a distribution function with all moments finite and such that the problem of moments for F has a nonunique solution ( F is M-indeterminate). Our goal is to explicitly describe a Stieltjes class S = { f ɛ = f [ 1 + ɛ h ] , ɛ ∈ [ - 1 , 1 ] } of distributions (here written in terms of the densities) all having the same moments as F. We study in detail the case when F is the distribution of the power transformation ξ r , r > 0 of a random variable ξ with generalized gamma distribution. We derive new Stieltjes classes in this case and also for powers of the normal and the exponential distributions. We find the value of the index of dissimilarity for some of these classes.

Stieltjes classes for moment-indeterminate probability distributions

LetFbe a probability distribution function with densityf. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments forFhas a nonunique solution (Fis M-indeterminate). Our goal is to describe a, wherehis a ‘small' perturbation function. Such a classSconsists of different distributionsFε(fεis the density ofFε) all sharing the same moments as those ofF, thus illustrating the nonuniqueness ofF, and of anyFε,in terms of the moments. Power transformations of distributions such as the normal, log-normal and exponential are considered and for them Stieltjes classes written explicitly. We define a characteristic ofScalled anindex of dissimilarityand calculate its value in some cases. A new Stieltjes class involving a power of the normal distribution is presented. An open question about the inverse Gaussian distribution is formulated. Related topics are briefly discussed.

Stieltjes classes for moment-indeterminate probability distributions

LetFbe a probability distribution function with densityf. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments forFhas a nonunique solution (Fis M-indeterminate). Our goal is to describe a, wherehis a ‘small' perturbation function. Such a classSconsists of different distributionsFε(fεis the density ofFε) all sharing the same moments as those ofF, thus illustrating the nonuniqueness ofF, and of anyFε,in terms of the moments. Power transformations of distributions such as the normal, log-normal and exponential are considered and for them Stieltjes classes written explicitly. We define a characteristic ofScalled anindex of dissimilarityand calculate its value in some cases. A new Stieltjes class involving a power of the normal distribution is presented. An open question about the inverse Gaussian distribution is formulated. Related topics are briefly discussed.

The Nevanlinna-Pick interpolation problems and power moment problems for matrix-valued functions

This paper is concerned with the study of several types of matrix interpolation problems, including • • The matrix Nevanlinna-Pick interpolation problem in the Nevanlinna class N p , • • The matrix Nevanlinna-Pick interpolation problem in the Stieltjes class L p , • • The truncated matrix power moment problem on the real axis, • • The Stieltjes matrix power moment problem. We use the so-called block Hankel vector approach for the treatment of these problems and make explicit the correspondence between solutions to the Nevanlinna-Pick interpolation problem and solutions to an associated power moment problem with the block Hankel vector under consideration. This enables one not only to find existence criteria for the solutions of the Nevanlinna-Pick interpolation problem, but also to describe the solutions based on the results of power moment problem, and vice versa.

The two-sided residue interpolation in the stieltjes class for matrix functions

The general two-sided directional interpolation problem in the Stieltjes class for matrix functions is solved. An explicit description of all solutions is presented in terms of the original data. The method is based on the reduction of this problem to the solution of homogeneous interpolation problems with symmetries for rationalmatrix functions.