Hamburger moment problems and orthogonal polynomials

Chihara, T. S.

doi:10.1090/s0002-9947-1989-0986686-1

Cited by 38 publications

(8 citation statements)

References 14 publications

(9 reference statements)

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“…We remark that in general, to find out whether a selfadjoint Jacobi matrix with exponentially growing entries is selfadjoint or has defect indices (1, 1), one can apply [11], Theorem 2 and its Corollary.…”

Section: Notice Thatmentioning

confidence: 99%

Completeness of rank one perturbations of normal operators with lacunary spectrum

Baranov

Yakubovich

2018

J. Spectr. Theory

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Suppose A is a compact normal operator on a Hilbert space H with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let L be its rank one perturbation. We show that either infinitely many moment equalities hold or the linear span of root vectors of L, corresponding to non-zero eigenvalues, is of finite codimension in H. In contrast to classical results, we do not assume the perturbation to be weak. M.S.C.(2000): Primary: 42A65; Secondary: 42C30 Keywords: selfadjoint operator, rank one perturbation, completeness of eigenvectors, Pólya peaks n |s n | −k | P n a, b | < ∞, but the equality (M k ) does not hold. Then L and L * are nearly complete.Moreover, for any ε > 0 there is a radius r > 0 such that the intersection of the non-zero spectrum σ(L) \ {0} with the disc B(0, r) is contained in the union of angles α ℓ − ε < arg z < α ℓ + ε, 1 ≤ ℓ ≤ n.

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Section: Notice Thatmentioning

confidence: 99%

Completeness of rank one perturbations of normal operators with lacunary spectrum

Baranov

Yakubovich

2018

J. Spectr. Theory

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“…. and in turn, it guarantees existence of a positive orthogonality measure on the real line −∞ < x < ∞ (so-called the Hamburger moment problem [26], [9]). If this measure is unique (up to a normalization condition) then the Hamburger moment problem is called the determinate.…”

Section: The Orthogonality Measure and Recurrence Coefficients For Th...mentioning

confidence: 99%

“…Finding criteria for determinacy of the Hamburger moment problem is a nontrivial problem [9]. However, there is a simple sufficient condition proposed by Carleman [26]: if…”

Section: The Orthogonality Measure and Recurrence Coefficients For Th...mentioning

confidence: 99%

Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials

Zhedanov

2013

Ramanujan J

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We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials which can be considered as a generalization of the Stieltjes-Carlitz elliptic polynomials. Relations between characteristic (i.e. positive definite) functions, Toda chain and orthogonal polynomials are developed in order to derive main properties of these polynomials. The recurrence coefficients and the weight function of these polynomials are expressed explicitly. In the degenerated cases of the elliptic functions the modified Meixner polynomials and the Krall-Laguerre polynomials appear.

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“…Three-term recurrence relations (2) and (3) provide useful information about the moment problem. For example, in [4], Chihara proved that, under the assumption…”

Section: Introductionmentioning

confidence: 99%