A Partial Characterization of Poised Hermite–Birkhoff Interpolation Problems

Atkinson, Kendall; Sharma, A.

doi:10.1137/0706021

Cited by 100 publications

(42 citation statements)

References 5 publications

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“…If the matrix E satisfies the Pólya condition and if it has only even blocks, then it follows from the Atkinson-Sharma theorem [1] that (3) D(x,E)¿0…”

Section: An Auxiliary Resultsmentioning

confidence: 99%

“…Then, by the Atkinson-Sharma theorem [1], the Birkhoff interpolation problem, based on (xa, Ea), is poised.…”

Section: The Main Comparison Resultsmentioning

confidence: 99%

“…Clearly, E has only even interior blocks, \E\ = a(E), and E satisfies the Pólya condition. Then, by the well-known Atkinson-Sharma theorem [1], there is a polynomial P(t) = f(E) + ■■■ such that PU)(x¡) = 0 if ¥¡j = 1. Since P(t) has a constant sign on [a, b], /fl*P(t)dt ^ 0.…”

Section: =0 Y'=0mentioning

confidence: 99%

“…Then E = E, and hence a(E) = \E\. Note that E is poised, hence, by the Atkinson-Sharma theorem [1], we can construct an interpolatory quadrature formula (1) based on E. By construction, ADP(l) = \E\ -1. Thus, in view of (2), for any Pólya matrix E with even interior blocks, the corresponding interpolatory quadrature formula ( 1 ) has maximal ADP.…”

Section: =0 Y'=0mentioning

confidence: 99%

“…Evidently, the quantity I fb R(x,E):=\ / Cl(x,E;t)dt I Ja is the major term in the estimate of the error of (1).…”

Section: =0 Y'=0mentioning

confidence: 99%

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Comparison of Birkhoff Type Quadrature Formulae

Bojanov¹,

Nikolov²

1990

Mathematics of Computation

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Abstract. The classical approach to the theory of quadrature formulae is based on the concept of algebraic degree of precision (ADP). A quadrature formula ß, is considered to be "better" than Q2 if ADP(g, ) > ADP(ß2). However, there are many quadratures that use the same number of evaluations of the integrand and have the same ADP. Then, how should one compare such formulae? We show in this paper that the error of the quadrature depends monotonically on the type of data used. Roughly speaking, the lower the order of the derivatives used, the smaller is the error.As a consequence of the main result we demonstrate the existence of BirkhofF quadrature formulae of double precision.

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“…If the matrix E satisfies the Pólya condition and if it has only even blocks, then it follows from the Atkinson-Sharma theorem [1] that (3) D(x,E)¿0…”

Section: An Auxiliary Resultsmentioning

confidence: 99%

“…Then, by the Atkinson-Sharma theorem [1], the Birkhoff interpolation problem, based on (xa, Ea), is poised.…”

Section: The Main Comparison Resultsmentioning

confidence: 99%

Section: =0 Y'=0mentioning

confidence: 99%

Section: =0 Y'=0mentioning

confidence: 99%

“…Evidently, the quantity I fb R(x,E):=\ / Cl(x,E;t)dt I Ja is the major term in the estimate of the error of (1).…”

Section: =0 Y'=0mentioning

confidence: 99%

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Comparison of Birkhoff Type Quadrature Formulae

Bojanov¹,

Nikolov²

1990

Mathematics of Computation

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Der Satz von Budan ‐ Fourier für allgemeine polynomiale Monosplines

Stieglitz

1979

Z Angew Math Mech

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Ist E = (eϱ) eine Inzidenzmatrix aus 0 und 1 und X eine feste Menge von Knoten χϱ, so gehört zu (E, X) die Menge der allgemeinen polynomialen Monosplines M vom Grad n, M(x):= \documentclass{article}\pagestyle{empty}\begin{document}$ M\left({x} \right)\,: = \frac{{x^n}}{{n!}} + \sum\limits_{v = 0}^{n - 1} {a_v x^v} + \sum\limits_{e_{\rho v} = 1} {a_{\rho v} \left({x - \xi _\rho} \right)_ + ^{n - 1 - v}} $\end{document}. Es wird zunächst eine obere Schranke für die Nullstellen von M im Intervall (0, 1) angegeben. Anwendungen betreffen dann solche Monosplines M, die aus Quadraturformeln entstehen, welche für Polynome vom Höchstgrad n—1 exakt sind. In diesem Fall muß eine zu E gehörige Matrix Ẽ der Pólya‐Bedingung genügen, und M hat im Intervall (0, 1) keine Nullstellen.

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Lacunary Quadrature Formulae and Interpolation Singularity

Dimitrov

1993

Journal of Approximation Theory

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A Partial Characterization of Poised Hermite–Birkhoff Interpolation Problems

Cited by 100 publications

References 5 publications

Comparison of Birkhoff Type Quadrature Formulae

Comparison of Birkhoff Type Quadrature Formulae

Der Satz von Budan ‐ Fourier für allgemeine polynomiale Monosplines

Lacunary Quadrature Formulae and Interpolation Singularity

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