Moments and Positivity

Demanze, Olivier

doi:10.1006/jmaa.2000.6882

Cited by 7 publications

(6 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that similar results in the scalar case have been obtained in ( [3], Theorem 3.5), as well as ( [11], Corollary 4.3).…”

Section: Vol 49 (2004)supporting

confidence: 66%

See 1 more Smart Citation

On Factorization of Trigonometric Polynomials

Dritschel

2004

Integral Equations and Operator Theory

View full text Add to dashboard Cite

We give a new proof of the operator version of the Fejér-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejér-Riesz factorization is obtained. The extremal case, where the outer factorization is also * -outer, is examined in greater detail. The connection to Agler's model theory for families of operators is considered, and a set of families lying between the numerical radius contractions and ordinary contractions is introduced. The methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over R m , m ≥ 1, in terms of rational functions with fixed denominators. Mathematics Subject Classification (2000). 47A68 (Primary), 47A65, 42A05, 30C10, 15A23 (Secondary).

show abstract

“…Note that similar results in the scalar case have been obtained in ( [3], Theorem 3.5), as well as ( [11], Corollary 4.3).…”

Section: Vol 49 (2004)supporting

confidence: 66%

“…Factorization using fixed denominators has been of more recent interest. The reader is referred, for example, to [12], [11] and [3]. The latter, in particular, considers factorizations where the denominators have the same form as those here.…”

Section: Vol 49 (2004)mentioning

confidence: 99%

On Factorization of Trigonometric Polynomials

Dritschel

2004

Integral Equations and Operator Theory

View full text Add to dashboard Cite

show abstract

“…A sequence {γ ij } 0≤i+j≤m satisfying (1.1) will be called a truncated moment sequence and the solution µ is said to be a representing measure associated to the sequence {γ ij } 0≤i+j≤m . In [34] J. Stochel has shown that solving TCMP solves the widely studied Full Moment Problem (see, for example, [1,2,3,17,29,30,33,36]). More precisely, a full moment sequence {γ ij } i,j∈Z+ admits a representing measure if and only if each of its truncation γ (m) admits a representing measure.…”

Section: Introductionmentioning

confidence: 99%

The quintic complex moment problem

Azhar¹,

Harrat²,

Idrissi³

et al. 2019

Operators and Matrices

View full text Add to dashboard Cite

Let γ (m) ≡ {γ ij } 0≤i+j≤m be a given complex-valued sequence. The truncated complex moment problem (TCMP in short) involves determining necessary and sufficient conditions for the existence of a positive Borel measure µ on C (called a representing measure for γ (m) ) such that γ ij = z i z j dµ for 0 ≤ i + j ≤ m. The TCMP has been completely solved only when m = 1, 2, 3, 4.We provide in this paper a concrete solution to the quintic TCMP (that is, when m = 5). We also study the cardinality of the minimal representing measure. Based on the bivariate recurrences sequences's properties with some Curto-Fialkow's results, our method intended to be useful for all odd-degree moment problems.2010 Mathematics Subject Classification. Primary 47A57, 15A83; Secondary 30E05, 44A60.

show abstract

“…In addition to its importance for applications, a complete solution of TMP would readily lead to a solution of the full moment problem, via a weak-* convergence argument, as shown by J. Stochel [40]. While we primarily focus on truncated moment problems, the full moment problem (in one or several variables) has been widely studied; see, for example, [1], [2], [17], [23], [24], [25], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [41], [42], [43].…”

Section: Introductionmentioning

confidence: 99%

Non-extremal sextic moment problems

Curto

Yoo

2015

Journal of Functional Analysis

View full text Add to dashboard Cite

Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems. Also, positive semidefiniteness, combined with another necessary condition, 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. However, these conditions are not sufficient for non-extremal sextic or higher-order truncated moment problems.In this paper we settle three key instances of the non-extremal (i.e., r < v) sextic moment problem, as follows: when r = 7, positive semidefiniteness, consistency and the variety condition guarantee the existence of a 7-atomic representing measure; when r = 8 we construct two determining algorithms, corresponding to the cases v = 9 and v = +∞. To accomplish this, we generalize the rank-reduction technique developed in previous work, where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure.

show abstract

Moments and Positivity

Abstract: We express the structure of some positive polynomials in several variables, as squares of rational function with universal denominators, using functional analysis methods. ᮊ

Cited by 7 publications

References 5 publications

On Factorization of Trigonometric Polynomials

On Factorization of Trigonometric Polynomials

The quintic complex moment problem

Non-extremal sextic moment problems

Contact Info

Product

Resources

About