The Nevanlinna matrix of the truncated Hausdorff matrix moment problem via orthogonal matrix polynomials on [a,b] for the case of an even number of moments

We consider the truncated Hausdorff matrix moment problem (THMM) in case of a finite number of even moments to be called non degenerate if two block Hankel matrices constructed via the moments are both positive definite matrices. The set of solutions of the THMM problem in case of a finite number of even moments is given with the help of the block matrices of the so-called resolvent matrix. The resolvent matrix of the THMM problem in the non degenerate case for matrix moments of dimension $q\times q$, is a $2q\times 2q$ matrix polynomial constructed via the given moments. In 2001, in [Yu.M. Dyukarev, A.E. Choque Rivero, Power moment problem on compact intervals, Mat. Sb.-2001. -69(1-2). -P.175-187], the resolvent matrix $V^{(2n+1)}$ for the mentioned THMM problem was proposed for the first time. In 2006, in [A. E. Choque Rivero, Y. M. Dyukarev, B. Fritzsche and B. Kirstein, A truncated matricial moment problem on a finite interval, Interpolation, Schur Functions and Moment Problems. Oper. Theory: Adv. Appl. -2006. - 165. - P. 121-173], another resolvent matrix $U^{(2n+1)}$ for the same problem was given. In this paper, we prove that there is an explicit relation between these two resolvent matrices of the form $V^{(2n+1)}=A U^{(2n+1)}B$, where $A$ and $B$ are constant matrices. We also focus on the following difference: For the definition of the resolvent matrix $V^{(2n+1)}$, one requires an additional condition when compared with the resolvent matrix $U^{(2n+1)}$ which only requires that two block Hankel matrices be positive definite. In 2015, 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, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259], a representation of the resolvent matrix of 2006 via matrix orthogonal polynomials was given. In this work, we do not relate the resolvent matrix $V^{(2n+1)}$ with the results of [A. E. Choque Rivero, From the Potapov to the Krein-Nudel'man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259]. The importance of the relation between $U^{(2n+1)}$ and $V^{(2n+1)}$ is explained by the fact that new relations among orthogonal matrix polynomials, Blaschke-Potapov factors, Dyukarev-Stieltjes parameters, and matrix continued fraction can be found. Although in the present work algebraic identities are used, to prove the relation between $U^{(2n+1)}$ and $V^{(2n+1)}$, the analytic justification of both resolvent matrices relies on the V.P. Potapov method. This approach was successfully developed in a number of works concerning interpolation matrix problems in the Nevanlinna class of functions and matrix moment problems.

We obtain a new multiplicative decomposition of the resolvent matrix of the non-degenerate truncated Hausdorff matrix moment (THMM) problem in the case of odd and even number of moments with the help of Dyukarev–Stieltjes matrix parameters (DSMP). Our result generalizes the Dyukarev representation of the resolvent matrix of the truncated Stieltjes matrix moment problem published in (Math Notes 75(1–2):66–82, 2004). In the scalar case, these parameters appear in the celebrated Stieltjes’s (1894) work Recherches sur les fractions continues and are used to establish the determinateness of the moment problem. We also obtain explicit relations between four families of orthogonal matrix polynomials on [a, b] together with their matrix polynomials of the second kind and the DSMP of the THMM problem. Additionally, we derive new representations of the Christoffel–Darboux kernel.

The multiplicative structure of the resolvent matrix of the Hausdorff Matrix Moment (HMM) problem is described in the case of an odd number of moments. We use the Fundamental Matrix Inequality approach, previously used in obtaining the Blaschke–Potapov product of the resolvent matrix for the Hamburger and Stieltjes matrix moment problem studied in [10] and [7], respectively. The case of an even number of moments for the HMM problem was considered in [12].

We obtain explicit interrelations between new Dyukarev-Stieltjes matrix parameters and orthogonal matrix polynomials on a finite interval [a, b], as well as the Schur complements of the block Hankel matrices constructed through the moments of the truncated Hausdorff matrix moment (THMM) problem in the nondegenerate case. Extremal solutions of the THMM problem are described with the help of matrix continued fractions.

In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form y = q ( x ) , q ( x ) ∈ R [ x ] , deg ⁡ q ≥ 3 and y x ℓ = 1 , ℓ ∈ N ∖ { 1 } . For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: 10.1007/s00020-021-02628-6.; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. http://doi.org/10.1016/j.laa.2022.05.008.; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: 10.1016/j.jmaa.2022.126563.], and improve Fialkow's bound to deg ⁡ q − 1 (resp. ℓ + 1 ) for curves y = q ( x ) (resp. y x ℓ = 1 ). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are kdeg ⁡ q (resp. k ( ℓ + 1 ) ) for curves y = q ( x ) (resp. y x ℓ = 1 ) for even degree sequences, while for odd ones they are kdeg ⁡ q − ⌈ deg ⁡ q 2 ⌉ (resp. k ( ℓ + 1 ) − ⌊ ℓ 2 ⌋ + 1 ) for curves y = q ( x ) (resp. y x ℓ = 1 ). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves y = x ℓ , ℓ = 2 , 3 , and add a new solvability condition to the even degree case on the curve y = x 2 .

Previous article Next article Conditions for Positive and Nonnegative Definiteness in Terms of PseudoinversesArthur AlbertArthur Alberthttps://doi.org/10.1137/0117041PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Richard Bellman, Introduction to matrix analysis, McGraw-Hill Book Co., Inc., New York, 1960xx+328 MR0122820 0124.01001 Google Scholar[2] A. Ben-Israel and , A. Charnes, Contributions to the theory of generalized inverses, J. Soc. Indust. Appl. Math., 11 (1963), 667–699 10.1137/0111051 MR0179192 0116.32202 LinkISIGoogle Scholar[3] T. N. E. Greville, Some applications of the pseudoinverse of a matrix, SIAM Rev., 2 (1960), 15–22 10.1137/1002004 MR0110185 0168.13303 LinkISIGoogle Scholar[4] Charles A. Rohde, Generalized inverses of partitioned matrices, J. Soc. Indust. Appl. 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The tracial moment problem on quadratic varieties

For a degree 2n finite sequence of real numbers \(\beta \equiv \beta ^{(2n)}= \{ \beta _{00},\beta _{10},\)\(\beta _{01},\ldots , \beta _{2n,0}, \beta _{2n-1,1},\ldots , \beta _{1,2n-1},\beta _{0,2n} \}\) to have a representing measure \(\mu \), it is necessary for the associated moment matrix \(\mathcal {M}(n)\) to be positive semidefinite, and for the algebraic variety associated to \(\beta \), \(\mathcal {V}_{\beta } \equiv \mathcal {V}(\mathcal {M}(n))\), to satisfy \({\text {rank}} \mathcal {M}(n)\le {\text {card}} \mathcal {V}_{\beta }\) as well as the following consistency condition: if a polynomial \(p(x,y)\equiv \sum _{ij}a_{ij}x^{i}y^j\) of degree at most 2n vanishes on \(\mathcal {V}_{\beta }\), then the Riesz functional\(\Lambda (p) \equiv p(\beta ):=\sum _{ij}a_{ij}\beta _{ij}=0\). Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic (\(n=1\)) and quartic (\(n=2\)) moment problems. Also, positive semidefiniteness, combined with consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r) and the cardinality of the associated algebraic variety (denoted by v) are equal. For extremal sextic moment problems, verifying consistency amounts to having good representation theorems for sextic polynomials in two variables vanishing on the algebraic variety of the moment sequence. We obtain such representation theorems using the Division Algorithm from algebraic geometry. As a consequence, we are able to complete the analysis of extremal sextic moment problems.

The four families of matrix orthogonal polynomials are considered arising in the truncated Hausdorff matrix moment (THMM) problem. Two of those families are associated with an odd number of moments and the other two with an even number of moments. The three-term recurrence relations associated with these four families are investigated. Certain explicit formulas are presented relating the three-term recurrence relation coefficients to the Dyukarev-Stieltjes parameters, the Schur complements and the orthogonal matrix polynomials associated with the THMM problem. The matrix version of the J-fraction is presented for the corresponding four extremal solutions of the THMM problem.

Truncated moment and related interpolation problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established and complete parametrizations of solutions are given. The framework used in the paper allows the treatment of even and odd order problems in a parallel manner. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new effects in the indefinite case is that the degenerate moment problem may have infinitely many solutions. However, with a careful application of an indefinite analogue of a step‐by‐step Schur algorithm a complete description of the set of solutions will be obtained.

In their book The Markov Moment Problem and Extremal Problems, published in 1977, M. G. Krein and A. A. Nudel’man presented a complete solution of the truncated Hausdorff moment problem via orthogonal polynomials on a finite interval [a, b]. By using the Potapov schema the matrix version of this moment problem was studied by the author, Yu. M. Dyukarev, B. Fritzsche and B. Kirstein. In the present work, we obtain the matrix generalisation of the above-mentioned Krein–Nudel’man representation. We also obtain explicit relations between four families of orthogonal matrix polynomials on [a, b] and their second kind polynomials, which are associated with the matrix version of the truncated Hausdorff moment problem.

Indeterminate symmetric moment problems

Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)j∈ℕn of real numbers and a closed subset F⊆ℝn, n∈{1,2,…}, find a positive regular Borel measure μ on F such that ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.

Non-extremal sextic moment problems

The truncated moment problem on the union of parallel lines