Orthogonal Laurent polynomials and the strong Hamburger moment problem

Jones, William B.; Thron, W. J.; Njåstad, Olav

doi:10.1016/0022-247x(84)90267-1

Cited by 98 publications

(77 citation statements)

References 6 publications

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“…They also stated the SHMP. These ideas were then further elaborateed by Jones and the present authors in [4] to show that the positive definiteness of the bilinear form associated with the sequence {c n : n = 0, +1, + 2,...} (see Section 1), or equivalently the determinant condition…”

Section: J Imentioning

confidence: 84%

“…We recall that two successive orthogonal L-polynomials Q n (z) and C + 1 ( z ) a r e n o t both singular; see Section 1 (cf. Corollary 3.4 of [4]). It follows from Theorem 4.3 that for every m, either A 2m (z) = E_ 2mi2m _ 2 (z) or A 2m+1 (z) = £_ 2m)2n ,(z).…”

Section: The Uniqueness Theoremmentioning

confidence: 95%

“…There will not be a unique solution to the SSMP if the positive T-fraction terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870002646X 8 Olav Njastad and W. J. Thron [4] determined by the moments diverges (see [9]). Hence for this sequence of moments the SHMP also must have an infinite number of solutions.…”

Section: J Imentioning

confidence: 99%

“…Hence for this sequence of moments the SHMP also must have an infinite number of solutions. In [4] we have exhibited a class of moment sequences for which the SHMP has a unique solution. Hence both alternatives are possible.…”

Section: J Imentioning

confidence: 99%

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Unique solvability of the strong Hamburger moment problem

Njåstad

Thron

1986

J Aust Math Soc A

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Methods from the theory of orthogonal polynomials are extended to L-polynomials Y. q n = p a n z". By this means the authors and W. B. Jones (J. Math. Anal. Appl. 98 (1984), 528-554) solved the strong Hamburger moment problem, that is, given a double sequence {c n }°? x , to find a distribution function i|/(r), non-decreasing, with an infinite number of points of increase and bounded on -oo < t < oo, such that for all integers n, c n = f™ x (-t) n dip(t). In this article further methods such as analogues of the Liouville-Ostrogradski formula and of the Christoffel-Darboux formula are developed to investigate when the moment problem has a unique solution. This will be the case if and only if a sequence of nested disks associated with the sequence { c n }™ x has only a point as its intersection (the so called limit point case).

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Section: J Imentioning

confidence: 84%

Section: The Uniqueness Theoremmentioning

confidence: 95%

Section: J Imentioning

confidence: 99%

Section: J Imentioning

confidence: 99%

See 2 more Smart Citations

Unique solvability of the strong Hamburger moment problem

Njåstad

Thron

1986

J Aust Math Soc A

Self Cite

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“…That is functions il/(t) which are bounded and non-decreasing in ( -00,00) and for which the moments n = \ are finite for n = 0, ±1, +2,.... Such functions have been described as strong distribution functions because they arise as solutions of strong moment problems (see [1,2]). The distribution is called symmetric if all the odd order moments are zero and is called a positive half distribution if all the points of increase are on the positive real axis.…”

Section: Introductionmentioning

confidence: 99%