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 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.

TIME OPTIMALITY AND THE POWER MOMENT PROBLEM

1989 ◽  
Vol 62 (1) ◽  
pp. 185-206 ◽  
Author(s):  
V I Korobov ◽  
G M Sklyar
Keyword(s):  
Moment Problem ◽  
Power Moment ◽  
Time Optimality ◽  
Power Moment Problem

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.

MONOTONE OPERATOR FUNCTIONS ON C*-ALGEBRAS

2005 ◽  
Vol 16 (02) ◽  
pp. 181-196 ◽  
Author(s):  
HIROYUKI OSAKA ◽  
SERGEI SILVESTROV ◽  
JUN TOMIYAMA
Keyword(s):  
Classical Case ◽  
Linear Operators ◽  
Monotone Functions ◽  
Bounded Linear ◽  
C Algebra ◽  
C Algebras ◽  
Operator Monotone ◽  
Operator Functions ◽  
Operator Monotone Functions

The article is devoted to investigation of classes of functions monotone as functions on general C*-algebras that are not necessarily the C*-algebra of all bounded linear operators on a Hilbert space as in classical case of matrix and operator monotone functions. We show that for general C*-algebras the classes of monotone functions coincide with the standard classes of matrix and operator monotone functions. For every class we give exact characterization of C*-algebras with this class of monotone functions, providing at the same time a monotonicity characterization of subhomogeneous C*-algebras. We use this result to generalize characterizations of commutativity of a C*-algebra based on monotonicity conditions for a single function to characterizations of subhomogeneity. As a C*-algebraic counterpart of standard matrix and operator monotone scaling, we investigate, by means of projective C*-algebras and relation lifting, the existence of C*-subalgebras of a given monotonicity class.

scholarly journals Large Deviations for random power moment problem

2004 ◽  
Vol 32 (3B) ◽  
pp. 2819-2837 ◽  
Author(s):  
Fabrice Gamboa ◽  
Li-Vang Lozada-Chang
Keyword(s):  
Large Deviations ◽  
Moment Problem ◽  
Power Moment ◽  
Power Moment Problem

scholarly journals A Potapov-Type Approach to a Truncated Matricial Stieltjes-Type Power Moment Problem

2020 ◽  
pp. 193-297
Author(s):  
Bernd Fritzsche ◽  
Bernd Kirstein ◽  
Conrad Mädler ◽  
Tatsiana Makarevich
Keyword(s):  
Moment Problem ◽  
Power Moment ◽  
Type Power ◽  
Power Moment Problem ◽  
Stieltjes Type
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