Binet’s formula for operator-valued recursive sequences and the operator moment problem

The scalar moment problem was first introduced by T. J. Stieltjes in his work ``Recherches sur les fractions continues'' Annals of the Faculty of Sciences of Toulouse 8, 1--122, (1895). He formulated it as follows: Given the moments of order $k$ ($k=0,1,2,\dots$), find a positive mass distribution on the half-line $[0,+\infty)$. The study of matrix and operator moment problems was initiated by M. G. Krein in his seminal paper ``Fundamental aspects of the representation theory of Hermitian operators with deficiency index $(m,m)$'' Translations of the American Mathematical Society, Series II, 97, 75--143, (1949). This paper is related to the truncated Hausdorff matrix moment (THMM) problem: the truncated moment problem on a compact interval $[a,b]$ in contrast to the Stieltjes moment problem on $[0,+\infty)$ and the Hamburger moment problem on $(-\infty,+\infty)$. Our approach relies on V. P. Potapov’s method, which reformulates interpolation and moment problems as equivalent matrix inequalities and introduces auxiliary matrices that satisfy the $\widetilde{J}_q$--inner function property of the Potapov class, together with a system of column pairs. The method begins by constructing Hankel matrices from the prescribed moments. If these matrices are positive semidefinite, the THMM problem is solvable. In the strictly positive definite case, known as the non-degenerate case, we transform the associated matrix inequalities to derive the Nevanlinna (or resolvent) matrix of the THMM problem, which characterizes its solutions. This framework has been extensively applied, for instance in A. E. Choque Rivero, Yu. M. Dyukarev, B. Fritzsche, and B. Kirstein, ``A truncated matricial moment problem on a finite interval'', in Interpolation, Schur Functions and Moment Problems, Operator Theory: Advances and Applications, Birkh\"{a}user, Basel, 165, 121--173, (2006). The main contribution of the present work is to represent the Nevanlinna matrix of the THMM problem in terms of orthogonal matrix polynomials (OMP) and their associated polynomials of the second kind at point $b$. Note that the representation at point $a$ was obtained earlier in A. E. Choque Rivero, ``From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem'' Bulletin of the Mexican Mathematical Society, 21(2), 233--259 (2015). In addition, we establish new identities involving OMP and reformulate an explicit relationship between the Nevanlinna matrices of the THMM problem at points $a$ and $b$, through OMP.

In this chapter, we study the determinacy problem in the multivariate case. In Sect. 14.1, we introduce several natural determinacy notions (strict determinacy, strong determinacy, ultradeterminacy) that are all equivalent to the “usual” determinacy in dimension one. In the remaining sections we develop various techniques and methods to derive sufficient criteria for determinacy. In Sect. 14.2, polynomial approximation is used to show that the determinacy of all marginal sequences implies the determinacy of a moment sequence (Theorem 14.6). Section 14.3 is based on operator-theoretic methods in Hilbert space. The main results (Theorems 14.12 and 14.16) show that the determinacy of appropriate 1-subsequences of a positive semidefinite d-sequence s implies that s is a (determinate) moment sequence. Section 14.4 is concerned with Carleman’s condition in the multivariate case. Probably the most useful result in this chapter is Theorem 14.20; it says that if all marginal sequences of a positive semidefinite d-sequence s satisfy Carleman’s condition, then s is a determinate moment sequence. Section 14.6 uses the disintegration of measures as a powerful method for the study of determinacy. A fibre theorem for determinacy (Theorem 14.30) states a measure is determinate if the base measure is strictly determinate and almost all fibre measures are determinate. In Sect. 14.5, we calculate the moments of the surface measure on S d−1 and of the Gaussian measure on \(\mathbb{R}^{d}\).

In this paper we develop a novel matrix method for solving linear recurrence relations and present explicit formulae for the general solution of the third-order linear homogeneous recurrence relations with variable coefficients. We obtain a summatory formula for the general solution of the recurrence relation in the special case. We review some known results and then consider some particular cases of the recurrence and examples with applications to combinatorics, especially to number sequences and polynomials. Finally, we briefly discuss further generalization of the method for higher order linear recurrence relations.

Given ℌ be Hilbert space over ℂ. If ℌ is a Hilbert space then ℌ2 is also Hilbert space. A linear relation on ℌ is a subspace of ℌ2. A linear relation can be multivalued part or not multivalued part. This paper proposes to discuss and show terms that a linear relation is a bounded linear operator and its spectrum analysis. We give the result that if ℜ is an injective relation on ℌ and range of ℜ is dense then ℜ* is also injective and (ℜ*)−1 = (ℜ−1)*. Consequently, if a relation ℜ on ℌ is an injective and self-adjoint, then a relation ℜ-1 is also self-adjoint. If a relation ℜ is a symmetric on ℌ then N(z - ℜ*) = R(z - ℜ)⊥∩ D(z - ℜ*) and N(z - ℜ) = R(z - ℜ*)⊥∩ D(z - ℜ). If a relation ℜ is a symmetric and z ∈ ℂ, then ∥za-b∥2 ≥ q2 ∥a∥2. If a relation ℜ is closed, bounded and |z| ≥ ∥b∥, then z ∈ ρ(ℜ). Consequently, if a relation ℜ is closed, bounded and |z| ≤ ∥b∥, then z ∈ σ(ℜ).

Results concerning some moment problems in unbounded sets from the scalar context to the case of linear operator data are extended. The present methods also lead to a theoretic characterization of a class of subnormal multioperators.

Unbounded extensions and operator moment problems

Moment methods in two point Padé approximation

Moment methods in Padé approximation

We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.

We present two algorithms to compute m-fold hypergeometric solutions of linear recurrence equations for the classical shift case and for the q-case, respectively. The first is an m-fold generalization and q-generalization of the algorithm by van Hoeij (Appl Algebra Eng Commun Comput 17:83–115, 2005; J. Pure Appl Algebra 139:109–131, 1998) for recurrence equations. The second is a combination of an improved version of the algorithms by Petkovšek (Discrete Math 180:3–22, 1998; J Symb Comput 14(2–3):243–264, 1992) for recurrence and q-recurrence equations and the m-fold algorithm from Petkovšek and Salvy (ISSAC 1993 Proceedings, pp 27–33, 1993) for recurrence equations. We will refer to the classical algorithms as van Hoeij or Petkovšek respectively. To formulate our ideas, we first need to introduce an adapted version of an m-fold Newton polygon and its characteristic polynomials for the classical case and q-case, and to prove the important properties in this case. Using the data from the Newton polygon, we are able to present efficient m-fold versions of the van Hoeij and Petkovšek algorithms for the classical shift case and for the q-case, respectively. Furthermore, we show how one can use the Newton polygon and our characteristic polynomials to conclude for which \({m\in \mathbb{N}}\) there might be an m-fold hypergeometric solution at all. Again by using the information obtained from the Newton polygon, the presentation of the q-Petkovšek algorithm can be simplified and streamlined. Finally, we give timings for the ‘classical’ q-Petkovšek, our q-van Hoeij and our modified q-Petkovšek algorithm on some classes of problems and we present a Maple implementation of the m-fold algorithms for the q-case.

Different types of methods have been used in runoff prediction involving conceptualand empirical models. Nevertheless, none of these methods can be considered as a single superior model. Owing to the complexity of the hydrological process, the accurate runoff is difficult to be predicted using the linear recurrence relations or physically based watershed. The linear recurrence relation model does not attempt to take into account the nonlinear dynamic of the hydrological process. The Artificial Neural Network (ANN) is a new technique with a flexible mathematical structure that is capable of identifying complex non-linear relationships between input and output data when compared to other classical modelling techniques. Therefore, the present study aims to utilize an Artificial Neural Network (ANN) to predict the rainfall-runoff relationship in a catchment area located in a Tanakami region of Japan. The study illustrates the applications of the feed forward back propagation with hyperbolic tangent neurons in the hidden layer and linear neuron in the output layer is used for rainfall prediction. To evaluate the performance of the proposed model, three statistical indexes were used, namely; Correlation coefficient (R), mean square error (MSE) and correlation of determination (R2). The results showed that the feed forward back propagation Neural Network (ANN) can describe the behaviour of rainfall-runoff relation more accurately than the classical regression model. Key words: Rainfall-runoff, artificial neural network, linear regression.

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of linear three-term recurrence relations, we show that there exists a four-term linear recurrence relation whose solutions show that the number has an irrational square if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher-order recurrence relations, and a transcendence criterion can also be formulated in terms of these principal solutions. The method generates new series expansions of positive integer powers of ζ(3) and ζ(2) in terms of Apéry’s now classic sequences.

Suppose / is a real valued function of bounded variation on [0,1]. Then for each nonnegative integer n, the Stieltjes integral 1 j n df exists, where for each number x, j(x) = x. A Jo necessary and sufficient condition is given for / in order that the moment sequence for /, {C n }n=o, is square summable. A second result establishes that the set of all such square summable moment sequences is dense in I 2 . LEMMA 1. If p is a number, 1/2 < p < 1, and for each nonnegatve integer n, a n = 1 -(n + l)~p then

One of the major unsolved problems in operator theory is the fifty-year-old invariant subspace problem, which asks whether every bounded linear operator on a Hilbert space has a nontrivial closed invariant subspace. This book presents some of the major results in the area, including many that were derived within the past few years and cannot be found in other books. Beginning with a preliminary chapter containing the necessary pure mathematical background, the authors present a variety of powerful techniques, including the use of the operator-valued Poisson kernel, various forms of the functional calculus, Hardy spaces, fixed point theorems, minimal vectors, universal operators and moment sequences. The subject is presented at a level accessible to postgraduate students, as well as established researchers. It will be of particular interest to those who study linear operators and also to those who work in other areas of pure mathematics.

The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials