Quasi orthogonal Jacobi polynomials and best one-sided <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si42.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> approximation to step functions

Bustamante, Jorge; Martínez-Cruz, Reinaldo; Quesada, José M.

doi:10.1016/j.jat.2015.05.001

Cited by 7 publications

(8 citation statements)

References 11 publications

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“…Let us outline only several exact results on problem (2.4) closely related to the present paper; for a more complete presentation of the topic see [1,11]. Problem (2.4) of one-sided integral approximation to the characteristic function of an arbitrary half-open interval (a, 1] ⊂ (−1, 1] by algebraic polynomials on [−1, 1] with the unit weight was solved, and the whole class of extremal polynomials was described in [5]. This problem in the space L υ (−1, 1) with an arbitrary weight is solved in [1].…”

Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 95%

“…in what follows, we sometimes will use more accurate (in comparison with (2.2)) notation for nodes is the left and right Radau quadrature formula, respectively; in the case u = {−1, 1}, (2.7) is the Lobatto quadrature formula. It is known (see the references in [1,3,5]) that formula (2.7) is positive in all these cases.…”

Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 96%

“…In this section, we give results from [1,5,11] on the one-sided approximation in the space L υ (−1, 1) to the characteristic function of an interval by algebraic polynomials. The results from Section 1 and from this sections make it possible to find the best one-sided approximation in the space L(S m−1 ) to the characteristic function of a spherical layer (in particular, a spherical cap) by algebraic polynomials in certain situations.…”

Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 99%

“…Problems of one-sided weighted integral approximation to the characteristic function of an interval and to similar functions by algebraic or trigonometric polynomials arise in various areas of mathematics and have a rich history (see [1,2,5,6,11,14,16,17] and the references therein). Let us outline only several exact results on problem (2.4) closely related to the present paper; for a more complete presentation of the topic see [1,11].…”

Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 99%

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One-Sided L-Approximation on a Sphere of the Characteristic Function of a Layer

Deikalova

Torgashova

2018

UMJ

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In the space L(S m−1) of functions integrable on the unit sphere S m−1 of the Euclidean space R m of dimension m ≥ 3, we discuss the problem of one-sided approximation to the characteristic function of a spherical layer G(J) = {x = (x 1 , x 2 ,. .. , xm) ∈ S m−1 : xm ∈ J}, where J is one of the intervals (a, 1], (a, b), and [−1, b), −1 < a < b < 1, by the set of algebraic polynomials of given degree n in m variables. This problem reduces to the one-dimensional problem of one-sided approximation in the space L φ (−1, 1) with the ultraspherical weight φ(t) = (1−t 2) α , α = (m−3)/2, to the characteristic function of the interval J. This result gives a solution of the problem of one-sided approximation to the characteristic function of a spherical layer in all cases when a solution of the corresponding one-dimensional problem known. In the present paper, we use results by A.G. Babenko, M.V. Deikalova, and Sz.G. Revesz (2015) and M.V. Deikalova and A.Yu. Torgashova (2018) on the one-sided approximation to the characteristic functions of intervals.

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Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 95%

Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 96%

Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 99%

Section: One-sided Approximation To the Characteristic Function Of Anmentioning

confidence: 99%

See 2 more Smart Citations

One-Sided L-Approximation on a Sphere of the Characteristic Function of a Layer

Deikalova

Torgashova

2018

UMJ

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“…In [6], Bustamante, Martínez-Cruz and Quesada apply the interlacing properties of zeros of quasi-orthogonal and orthogonal Jacobi polynomials given in [12] and in Lemma 1 to show that best possible one-sided polynomial approximants to a unit step function on the interval [−1, 1], which are in some cases unique, can be obtained using Hermite interpolation at interlaced zeros of quasi-orthogonal and orthogonal Jacobi polynomials.…”

Section: Introductionmentioning

confidence: 99%

Zeros of Quasi-Orthogonal Jacobi Polynomials

Driver

Jordaan

2016

SIGMA

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Abstract. We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by α > −1, −2 < β < −1. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials P (α,β) n and P (α,β+2) n are interlacing, holds when the parameters α and β are in the range α > −1 and −2 < β < −1. We prove that the zeros of P (α,β) n and P (α,β) n+1 do not interlace for any n ∈ N, n ≥ 2 and any fixed α, β with α > −1, −2 < β < −1. The interlacing of zeros of P (α,β) n and P (α,β+t) m for m, n ∈ N is discussed for α and β in this range, t ≥ 1, and new upper and lower bounds are derived for the zero of P (α,β) n that is less than −1.

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