On best error bounds for approximation by piecewise polynomial functions

Olof Widlund

Research output: Contribution to journalArticle

Abstract

An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.

Original languageEnglish (US)
Pages (from-to)327-338
Number of pages12
JournalNumerische Mathematik
Volume27
Issue number3
DOIs
StatePublished - Sep 1976

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Piecewise Polynomials
Polynomial function
Best Approximation
Error Bounds
Polynomials
Inverse Theorems
Polynomial Splines
Trigonometric Polynomial
Approximation
Hermite
Theorem
Approximation Methods
Smooth function
Interpolate
Mesh
Finite Element
Analogue
Splines
Interpolation
Family

Keywords

  • AMS Subject Classifications: 41A15, 41A25, 41A40, 41A65, 46E35

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • Mathematics(all)

Cite this

On best error bounds for approximation by piecewise polynomial functions. / Widlund, Olof.

In: Numerische Mathematik, Vol. 27, No. 3, 09.1976, p. 327-338.

Research output: Contribution to journalArticle

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