On the Application of the Peano Representation of Linear Functionals in Numerical Analysis

Brass, Helmut; Förster, Klaus-Jürgen

doi:10.1007/978-94-015-9086-0_10

Cited by 11 publications

(5 citation statements)

References 21 publications

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“…Here, deg(Q/J > s -1 means that for every ieR and every polynomial ρ of degree < s -1 we have Qh\p](x) = p(x)· The essential requirement in (1) is that all basis functions have a common majorant. An important problem for such approximations is the estimation of the error.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Error Bounds for Linear Approximations on the Real Line

Ehrich¹

2000

Analysis

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On the real line, several known error bounds for quasi-interpolants with slowly decaying basis functions do not yield the full order of approximation of the underlying approximation space. On the other hand, the unimprovability of these bounds is only known in some special cases. Developping new Peano kernel techniques for such operators, new pointwise error bounds as well as general unimprovability results are given. MATHEMATICS SUBJECT CLASSIFICATION: 41A80, 65D99 INTRODUCTION AND STATEMENT OF THE MAIN RESULTSWe consider quasi-interpolants of the type with Λ > 0, η € Ν, 0 < ε < 1, C independent of k and t, deg(Qh) > s -1. Here, deg(Q/J > s -1 means that for every ieR and every polynomial ρ of degree < s -1 we have Qh\p](x) = p(x)· The essential requirement in (1) is that all basis functions have a common majorant. An important problem for such approximations is the estimation of the error. In (1), the quasiinterpolants are based on localising basis functions, and their approximation properties depend on the local smoothness of /. Hence, pointwise error bounds are particularly interesting. In this paper, we develop new Peano kernel techniques for quasi-interpolants on the real line, in particular with slowly decaying basis functions. Using this approach, we derive new pointwise error bounds and unimprovability results. For s < η -1 in (1), the basis functions are sufficiently fast decaying such that the following Peano representation of the error functional can be stated for the standard class of functions (1) ke ζ ¥>/b,fc(t) =

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…We can apply (8) and obtain, with C\,C 2 independent of h, ΛΓ > Ci (l + h-^t-x))' 1 "~1 (2 + y + fc)" 71 " 1 . o<k<Y Let C > 0 be given, and let first 0 < y < Then (11) Taking the infimum of this constant and the one for 0 < y < the result follows for Κ = 0.…”

Section: \ <1mentioning

confidence: 99%

Error Bounds for Linear Approximations on the Real Line

Ehrich¹

2000

Analysis

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“…1,ν g (s) 2,s−ν for s = 6 and s = 7, we have the existence of a constant for x ∈ [7/16 · 2 −µ , 1/2 · 2 −µ ] depending on µ and s, but not depending on x, such that const · K 2s−2−4 (R Ro(s) ns,B )(x) ≤ −2g (6) 1,1 g (6) 2,5ř ( 1 2 ) cos(4 · 2 µ πx) + 2g (6) 1,2 g…”

Section: Proof Of Lemmamentioning

confidence: 95%

“…2,4ř ( 1 2 ) cos(6 · 2 µ πx) +2g (6) 1,0 g (6) 2,6ř ( 3 2 ) cos(3 · 2 µ πx) − 2g < −7 cos(4 · 2 µ πx) + 13 cos(6 · 2 µ πx) + cos(3 · 2 µ πx) −5 cos(8 · 2 µ πx) + 9 < 0 holds for µ = 0, 1, . .…”

Section: Proof Of Lemmaunclassified

Romberg quadrature using the Bulirsch sequence

Fischer

2002

Numerische Mathematik

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“…An important property of a quadrature formula is its positivity, i.e., the positivity of its coefficients a ν (in our case a [k] ν ). From theory, many nice properties follow from this positivity (see, e.g., Brass and Förster [9]). The coefficients are given by…”

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confidence: 95%

Multiple Precision Interval Packages: Comparing Different Approaches

Grimmer

Petras

Revol

2004

Numerical Software With Result Verification

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We give a survey on packages for multiple precision interval arithmetic, with the main focus on three specific packages. One is within a Maple environment, intpakX, and two are C/C++ libraries, GMP-XSC and MPFI. We discuss their different features, present timing results and show several applications from various fields, where high precision intervals are fundamental. Keywords:Multiple precision interval arithmetic, packages, ease of use, efficiency, reliability. RésuméDans cet article est effectué un tour d'horizon des outils implémentant l'arithmétique par intervalles en précision multiple. Un coup de projecteur est donné sur trois de ces outils. Le premier est un paquetage développé pour Maple: intpakX, les deux autres sont des bibliothèques C/C++: GMP-XSC et MPFI. Leurs spécificités sont présentées, puis leurs performances sont données. Enfin, sont développées quelques applications, issues de domaines d'application divers, pour lesquelles l'arithmétique par intervallesà grande précision est fondamentale.

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On the Application of the Peano Representation of Linear Functionals in Numerical Analysis

Cited by 11 publications

References 21 publications

Error Bounds for Linear Approximations on the Real Line

Error Bounds for Linear Approximations on the Real Line

Romberg quadrature using the Bulirsch sequence

Multiple Precision Interval Packages: Comparing Different Approaches

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