Exponentially bounded positive definite functions

Berg, Christian; Maserick, P. H.

doi:10.1215/ijm/1256046160

Cited by 48 publications

(45 citation statements)

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“…Atzmon [1] characterized the moment matrices associated with measures in the closed unit disk D = {z ∈ C: |z| 1} in terms of a property for a bi-sequence {c i, j } ∞ i, j=0 to be positive definite. The complex moment problem when a representing measure with compact support exists is completely characterized in [17] and in a general theorem of Berg and Maserick [4]. See also the treatment in [3].…”

Section: Introductionmentioning

confidence: 99%

Small eigenvalues of large Hermitian moment matrices

Escribano

Gonzalo

Torrano

2011

Journal of Mathematical Analysis and Applications

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We consider an infinite Hermitian positive definite matrix M which is the moment matrix associated with a measure μ with infinite and compact support on the complex plane.We prove that if the polynomials are dense in L 2 (μ) then the smallest eigenvalue λ n of the truncated matrix M n of M of size (n + 1) × (n + 1) tends to zero when n tends to infinity. In the case of measures in the closed unit disk we obtain some related results.

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Section: Introductionmentioning

confidence: 99%

Small eigenvalues of large Hermitian moment matrices

Escribano

Gonzalo

Torrano

2011

Journal of Mathematical Analysis and Applications

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“…For more information concerning the characterization of HPD matrices which are moment matrices with respect to a certain measure μ with support on C see among others [2,7,18]. Since moment matrices associated with measures with infinite support on C are HPD matrices, many of the examples that appear in this paper are indeed moment matrices.…”

Section: Introductionmentioning

confidence: 95%

On the inversion of infinite moment matrices

Escribano

Gonzalo

Torrano

2015

Linear Algebra and its Applications

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Motivated by [8] we study the existence of the inverse of an infinite Hermitian positive definite matrix (in short, HPD matrix) from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections. We prove a sufficient condition to assure the inversion of an HPD matrix with square summable rows. For infinite Toeplitz matrices we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of an infinite Toeplitz positive definite matrix is asymptotic Toeplitz. Such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials. We apply these results in the context of the theory of orthogonal polynomials. In particular, we show that for measures on the unit circle T verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures.

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“…These results have been revisited in [15] with a different approach, and were recently generalized in [7] to weighted p -norms, p 1. In [8] it is shown that the general result in [5] carries to the even smaller cone of sums of 2d-powers, R[X] 2d ⊆ R[X] 2 , where d 1 is an integer. In particular, R[X] 2d · 1 = Psd([−1, 1] n ).…”

Section: Introductionmentioning

confidence: 99%

Closure of the cone of sums of 2d-powers in commutative real topological algebras

Ghasemi¹,

Kuhlmann²

2013

Journal of Functional Analysis

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Let R be a unitary commutative R-algebra and K ⊆ X (R) = Hom(R, R), closed with respect to the product topology. We consider R endowed with the topology T (K), induced by the family of seminorms ρ α (a) := |α(a)|, for α ∈ K and a ∈ R. In case K is compact, we also consider the topology induced by a K := sup α∈K |α(a)| for a ∈ R. If K is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, R 2d , with respect to those two topologies is equal to Psd(K) := {a ∈ R: α(a) 0, for all α ∈ K}. In particular, any continuous linear functional L on the polynomial ring R = R[X] = R[X 1 , . . . , X n ] with L(h 2d ) 0 for each h ∈ R[X] is integration with respect to a positive Borel measure supported on K. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies.

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Exponentially bounded positive definite functions

Cited by 48 publications

References 1 publication

Small eigenvalues of large Hermitian moment matrices

Small eigenvalues of large Hermitian moment matrices

On the inversion of infinite moment matrices

Closure of the cone of sums of 2d-powers in commutative real topological algebras

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