scholarly journals On the Truncated Multidimensional Moment Problems in Cn

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 20
Author(s):  
Sergey Zagorodnyuk
Keyword(s):  
Integral Representation ◽  
Finite Subset ◽  
Moment Problem ◽  
Complex Numbers ◽  
Moment Problems ◽  
Linear Functionals ◽  
Truncated Moment Problem ◽  
The Moment ◽  
Multidimensional Moment Problem ◽  
Multidimensional Moment Problems

We consider the problem of finding a (non-negative) measure μ on B(Cn) such that ∫Cnzkdμ(z)=sk, ∀k∈K. Here, K is an arbitrary finite subset of Z+n, which contains (0,…,0), and sk are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn. Secondly, the moment problem is a particular case of the truncated moment problem in Cn, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials.

scholarly journals New Results on Markov Moment Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Octav Olteanu
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Moment Problems ◽  
Finite Dimensional ◽  
Integrable Functions ◽  
Truncated Moment Problem ◽  
Markov Moment ◽  
The Moment ◽  
Necessary And Sufficient

The present work deals with the existence of the solutions of some Markov moment problems. Necessary conditions, as well as necessary and sufficient conditions, are discussed. One recalls the background containing applications of extension results of linear operators with two constraints to the moment problem and approximation by polynomials on unbounded closed finite-dimensional subsets. Two domain spaces are considered: spaces of absolute integrable functions and spaces of analytic functions. Operator valued moment problems are solved in the latter case. In this paper, there is a section that contains new results, making the connection to some other topics: bang-bang principle, truncated moment problem, weak compactness, and convergence. Finally, a general independent statement with respect to polynomials is discussed.

Solutions of the Al-Salam–Chihara and allied moment problems

2019 ◽  
Vol 18 (02) ◽  
pp. 185-210 ◽  
Author(s):  
Mourad E. H. Ismail
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Infinite Family ◽  
Weight Functions ◽  
Moment Problems ◽  
Order Operator ◽  
Absolutely Continuous ◽  
Rodrigues Formula ◽  
Wilson Operator ◽  
The Moment

We study the moment problem associated with the Al-Salam–Chihara polynomials in some detail providing raising (creation) and lowering (annihilation) operators, Rodrigues formula, and a second-order operator equation involving the Askey–Wilson operator. A new infinite family of weight functions is also given. Sufficient conditions for functions to be weight functions for the [Formula: see text]-Hermite, [Formula: see text]-Laguerre and Stieltjes–Wigert polynomials are established and used to give new infinite families of absolutely continuous orthogonality measures for each of these polynomials.

scholarly journals The moment problem for continuous positive semidefinite linear functionals

2012 ◽  
Vol 100 (1) ◽  
pp. 43-53 ◽  
Author(s):  
Mehdi Ghasemi ◽  
Salma Kuhlmann ◽  
Ebrahim Samei
Keyword(s):  
Moment Problem ◽  
Positive Semidefinite ◽  
Linear Functionals ◽  
The Moment

scholarly journals The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

2021 ◽  
Author(s):  
Philipp J. di Dio ◽  
Mario Kummer
Keyword(s):  
Moment Problem ◽  
Hankel Matrix ◽  
Upper Bounds ◽  
Moment Problems ◽  
Algebraic Varieties ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
Flat Extension ◽  
Truncated Moment Problem ◽  
Carathéodory Number

AbstractIn this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $$\mathbb {R}^n$$ R n , and $$[0,1]^n$$ [ 0 , 1 ] n . We also treat moment problems with small gaps. We find that for every $$\varepsilon >0$$ ε > 0 and $$d\in \mathbb {N}$$ d ∈ N there is a $$n\in \mathbb {N}$$ n ∈ N such that we can construct a moment functional $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ d → R which needs at least $$(1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) $$ ( 1 - ε ) · n + d n atoms $$l_{x_i}$$ l x i . Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ 2 d → R which need to be extended to the worst case degree 4d, $$\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}$$ L ~ : R [ x 1 , ⋯ , x n ] ≤ 4 d → R , in order to have a flat extension.

scholarly journals MAXIMUM ENTROPY FOR REDUCED MOMENT PROBLEMS

2000 ◽  
Vol 10 (07) ◽  
pp. 1001-1025 ◽  
Author(s):  
MICHAEL JUNK
Keyword(s):  
Maximum Entropy ◽  
Wide Class ◽  
Moment Problem ◽  
Entropy Solution ◽  
Unbounded Domains ◽  
Entropy Solutions ◽  
Moment Problems ◽  
Precise Condition ◽  
The Moment

The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of ℝd is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.

scholarly journals The operator approach to the truncated multidimensional moment problem

2019 ◽  
Vol 6 (1) ◽  
pp. 1-19
Author(s):  
Sergey M. Zagorodnyuk
Keyword(s):  
Moment Problem ◽  
Dimensional Stability ◽  
General Type ◽  
Sufficient Conditions ◽  
Operator Approach ◽  
Numerical Examples ◽  
Associated Operators ◽  
The Moment ◽  
The Given ◽  
Multidimensional Moment Problem

Abstract We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. The case where the associated operators form a commuting self-adjoint tuple is characterized in terms of the given moments. The case of the dimensional stability is characterized in terms of the prescribed moments as well. Some sufficient conditions for the solvability of the moment problem are presented. A construction of the corresponding solution is described by algorithms. Numerical examples of the construction are provided.

scholarly journals The Stieltjes condition and multidimensional $${\mathcal {K}}$$-moment problems

2021 ◽  
Author(s):  
Konrad Schmüdgen
Keyword(s):  
Moment Problem ◽  
Moment Problems ◽  
Solvability Theorem ◽  
The Moment

AbstractWe prove a solvability theorem for the Stieltjes problem on $$\mathbb {R}^d$$ R d which is based on the multivariate Stieltjes condition $$\sum _{n=1}^\infty L(x_j^{n})^{-1/(2n)} =+\infty $$ ∑ n = 1 ∞ L ( x j n ) - 1 / ( 2 n ) = + ∞ , $$j=1,\dots ,d.$$ j = 1 , ⋯ , d . This result is applied to derive a new solvability theorem for the moment problem on unbounded semi-algebraic subsets of $$\mathbb {R}^d$$ R d .

scholarly journals Monotone operator functions, gaps and power moment problem

2007 ◽  
Vol 100 (1) ◽  
pp. 161 ◽  
Author(s):  
Hiroyuki Osaka ◽  
Sergei Silvestrov ◽  
Jun Tomiyama
Keyword(s):  
Moment Problem ◽  
Moment Problems ◽  
Monotone Functions ◽  
Power Moment ◽  
Hankel Matrices ◽  
Matrix Algebras ◽  
Truncated Moment Problem ◽  
Operator Functions ◽  
Operator Monotone Functions ◽  
Power Moment Problem

The article is devoted to investigation of the classes of functions belonging to the gaps between classes $P_{n+1}(I)$ and $P_{n}(I)$ of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps $P_{n}(I) \setminus P_{n+1}(I)$ for bounded intervals $I$. We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many polynomials in the gaps. We also provide via several examples some first insights into the further problem of description of polynomials in the gaps that are not coming from the truncated moment problem. Also, in this article, we deepen further in another way into the structure of the classes of matrix monotone functions and of the gaps between them by considering the problem of position in the gaps of certain interesting subclasses of matrix monotone functions that appeared in connection to interpolation of spaces and in a proof of the Löwner theorem on integral representation of operator monotone functions.

scholarly journals MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE

2020 ◽  
Vol 6 (1) ◽  
pp. 168
Author(s):  
Elias Zikkos
Keyword(s):  
Fourier Series ◽  
Entire Function ◽  
Real Line ◽  
Moment Problem ◽  
Series Representation ◽  
Growth Conditions ◽  
Complex Numbers ◽  
Moment Problems ◽  
Exponential System ◽  
Closed Span

For a class of sets with multiple terms$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},\underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots,\underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\},$$having density \(d\) counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak \(\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots ,\mu_n-1\} \) satisfying certain growth conditions, we consider a moment problem of the form $$\int_{-\infty}^{\infty}e^{-2w(t)}t^k e^{\lambda_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,\dots, \mu_n-1,$$ in weighted \(L^2 (-\infty, \infty)\) spaces. We obtain a solution \(f\) which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation $$ f(z)=\sum_{n=1}^{\infty}\Big(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\Big) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,\, z\in \mathbb{C}. $$ The proof depends on our previous work where we characterized the closed span of the exponential system \(\{t^k e^{\lambda_n t}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,\mu_n-1\}\) in weighted \(L^2 (-\infty, \infty)\) spaces, and also derived a sharp upper bound for the norm of elements of a biorthogonal sequence to the exponential system. The proof also utilizes notions from Non-Harmonic Fourier series such as Bessel and Riesz–Fischer sequences. 

scholarly journals On the Moment Problem and Related Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2289
Author(s):  
Octav Olteanu
Keyword(s):  
Moment Problem ◽  
Borel Measure ◽  
Truncated Moment Problem ◽  
Regular Borel Measure ◽  
The Subject ◽  
Markov Moment ◽  
The Moment ◽  
Adjoint Operators ◽  
Classical Moment Problem ◽  
Unknown Solution

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.

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