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1.
Given real numbers
with 00 >0 , the truncated parabolic moment problem for entails finding necessary and sufficient conditions for the existence of a positive Borel measure , supported in the parabola p(x, y) = 0, such that
We prove that admits a representing measure (as above) if and only if the associated moment matrix
is positive semidefinite, recursively generated and has a column relation p(X, Y) = 0, and the algebraic variety () associated to satisfies card
In this case, admits a rank
-atomic (minimal) representing measure.Submitted: August 25, 2003 相似文献
2.
Raúl E. Curto Lawrence A. Fialkow H. Michael Möller 《Integral Equations and Operator Theory》2008,60(2):177-200
For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness
of .
The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named
author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138. 相似文献
3.
For the quadratic complex moment problem
, we obtain necessary and sufficient conditions for the existence of representing measures supported in the unit circleT or in the closed unit disk
. We explicitly construct all finitely atomic representing measures supported inT or
which have the fewest atoms possible. For the quadratic
-moment problem in which the moment matrixM(1) is positive and invertible, there exists an ellipse D such that the minimal (3-atomic) representing measures are supported in the complement of the interior region of . Finally, we apply these results to obtain information on the location of the zeros of certain cubic polynomials.Dedicated to the memory of Velaho D. Bowman-FialkowBoth authors were partially supported by research grants from the National Science Foundation. The second-named author was also partially supported by an award from the State of New York/UUP Professional Development and Quality of Working Life Committee. 相似文献
4.
This paper is concerned with the solution of a certain tangential
Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called
block Hankel vector method to establish two intrinsic connections between
the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the
truncated Hamburger matrix moment problem associated with the block Hankel
vector under consideration: one is a congruent relationship between their
information matrices, and the other is a divisor-remainder connection between
their solutions. These investigations generalize our previous work on
the Nevanlinna-Pick interpolation and power matrix moment problem. 相似文献
5.
Both authors were partially supported by grants from NSF Raúl Curto was partially supported by a University of Iowa faculty scholar award 相似文献
6.
Both authors were partially supported by grants from NSF Raúl Curto was partially supported by a University of Iowa faculty scholar award 相似文献
7.
Raúl E. Curto Lawrence A. Fialkow H. Michael Möller 《Integral Equations and Operator Theory》2008,61(1):147-148
Due to a technical problem, we accidentally omitted a paragraph from the proof of one of the main results in [1]. In this
Addendum we provide the portion of the proof that did not appear in [1].
This is an addendum to . 相似文献
8.
The Operator Valued Autoregressive Filter Problem and the Suboptimal Nehari Problem in Two Variables
Necessary and sufficient conditions are given for the solvability of the operator valued two-variable autoregressive filter
problem. In addition, in the two variable suboptimal Nehari problem sufficient conditions are given for when a strictly contractive
little Hankel has a strictly contractive symbol. 相似文献
9.
We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure , the bound is based on estimating
where C:=C# [ ] is a positive matrix naturally associated with the moments of . We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag(c1,...,cn) (including the case when is planar measure on the unit disk), (C) is at least as large as the number of gaps ck >ck+1. 相似文献
10.
Hugo J. Woerdeman 《Integral Equations and Operator Theory》2002,42(2):229-242
The positive Carathéodory interpolation problem in the Agler-Herglotz class on the polydisc is solved, along with a several variable version of the Naimark dilation theorem. In addition, the positive Carathéodory interpolation problem for general holomorphic functions is discussed and numerical results are presented. 相似文献
11.
12.
The Generalized Moment Problem with Complexity Constraint 总被引:1,自引:0,他引:1
In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which
begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly
convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain
finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem.
Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of
interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we
give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications.
While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability
and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because
of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than
positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this
theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable
function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples
illustrating our results. 相似文献
13.
This paper presents a new version of the abstract band method. The new scheme applies to extension problems for classes of essentially bounded functions, continuous functions, and bounded operators, which were not covered by earlier versions of the abstract band method. 相似文献
14.
In this paper we shall show that the value of the maximum -entropy solution of an interpolation problem of the Nevanlinna-Pick type at the point maximizes the determinant of the solution of an associated matrix completion problem. This serves to show that the solutions of two distinct extremal problems coincide.Harry Dym wishes to express his thanks to Renee' and Jay Weiss for endowing the chair which supports his research. 相似文献
15.
16.
Toufik Mansour 《Discrete Mathematics》2006,306(12):1161-1176
We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. 相似文献
17.
Galliano Valent 《Mediterranean Journal of Mathematics》2006,3(2):327-345
In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the
spectral measure is under control of the polynomials asymptotics.
The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are
to be constructed using Nevanlinna parametrization. Nevertheless it is interesting to observe that some spectral measures
can still be obtained from weaker forms of the Markov theorem.
The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples
due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples. 相似文献
18.
19.
We study the connection between orthogonal polynomials in several variables and families of commuting symmetric operators of a special form. 相似文献
20.
Jagjit Singh Matharu Mohammad Sal Moslehian Jaspal Singh Aujla 《Linear algebra and its applications》2011,435(2):270-276
We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohr’s inequality due to Vasi? and Ke?ki?. 相似文献