On orthogonal polynomials

Geronimus, J.

doi:10.1090/s0002-9947-1931-1501591-0

Cited by 13 publications

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“…It is well known (polynomials orthogonal on the unit circle are studied extensively in [3]) that the Pn 's satisfy a recurrence relation of the type…”

Section: Preliminary Resultsmentioning

confidence: 99%

Linear combinations of orthogonal polynomials generating positive quadrature formulas

Peherstorfer¹

1990

Math. Comp.

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Abstract. Let pk(x) = x +■■■ , k e N0 , be the polynomials orthogonal on [-1, +1] with respect to the positive measure da . We give sufficient conditions on the real numbers p , j = 0, ... , m , such that the linear combination of orthogonal polynomials YfLo^jPn-j has n simple zeros in (-1,-1-1) and that the interpolatory quadrature formula whose nodes are the zeros of Yfj=oßjPn-j has positive weights.

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“…It is well known (polynomials orthogonal on the unit circle are studied extensively in [3]) that the Pn 's satisfy a recurrence relation of the type…”

Section: Preliminary Resultsmentioning

confidence: 99%

Linear combinations of orthogonal polynomials generating positive quadrature formulas

Peherstorfer¹

1990

Math. Comp.

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“…For every Borel positive and finite measure µ on [0, 2π ] it is very well known (see [6]) that the sequence of the norms of the SMOP is decreasing. [6,10]). If µ ∈ S, then the Szegö function is…”

Section: Orthogonality With Respect Toμ = µ + ν With ν a Bernstein-smentioning

confidence: 99%

“…. , is a sequence of positive Borel measures supported on the unit circle such that ν is the * -weak limit of the sequence {ν k } [6].…”

Section: Introductionmentioning

confidence: 99%

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

2006

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In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0, 2π] and a Bernstein-Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein-Szegö polynomials. When the Bernstein-Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.

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“…It is known that, when the reflection coefficients are different from zero, the Szegő polynomials and their reciprocals satisfy three term recurrence relation of the form (3.1). See for example, p. 91 of [7] and p. 60 and Theorem 1.6.11 of [18]. For a contribution, where M is represented by an integral on the unit circle, but is such that Q n (0) = 1, n ≥ 1, we refer to [4].…”

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confidence: 99%

A Characterization of L-orthogonal Polynomials from Three Term Recurrence Relations

et al. 2010

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We consider the sequence of polynomials {Q n } satisfying the L-orthogonality M[z −n+m Q n (z)] = 0, 0 ≤ m ≤ n − 1, with respect to a linear functional M for which the moments M[t n ] = μ n are all complex. Under certain restriction on the moment functional these polynomials also satisfy a three term recurrence relation. We consider three special classes of such moment functionals and characterize them in terms of the coefficients of the associated three term recurrence relations. Relations between the polynomials {Q n } associated with two of these special classes of moment functionals are also given. Examples are provided to justify this characterization.

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On orthogonal polynomials

Cited by 13 publications

References 0 publications

Linear combinations of orthogonal polynomials generating positive quadrature formulas

Linear combinations of orthogonal polynomials generating positive quadrature formulas

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

A Characterization of L-orthogonal Polynomials from Three Term Recurrence Relations

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