On Approximation and Interpolation of Convex Functions

Marian Neamtu

doi:10.1007/978-94-011-2634-2_28

On Approximation and Interpolation of Convex Functions

Marian Neamtu

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

Abstract

Some negative results concerning the convexity preserving approximation and interpolation of multivariate functions are presented. We prove that the approximation based on both interpolation and local operators cannot be convexity preserving, provided the approximation space is (locally) finite dimensional. In both cases we can dispense with the asssumption of the linearity of the approximation operator and the assumption that the approximation space is a space of piecewise polynomials. Some consequences for the construction of shape preserving approximations are discussed.

Original language	English
Title of host publication	Approximation Theory, Spline Functions and Applications
Editors	S.P. Singh
Place of Publication	Dordrecht
Publisher	Springer
Pages	411–418
ISBN (Electronic)	978-94-011-2634-2
ISBN (Print)	978-0-7923-1574-2, 978-94-010-5164-4
DOIs	https://doi.org/10.1007/978-94-011-2634-2_28
Publication status	Published - 1992

Publication series

Name	Nato Science Series C
Publisher	Springer
Volume	356
ISSN (Print)	1389-2185

Access to Document

10.1007/978-94-011-2634-2_28

1 Report

On Approximation and interpolation of Convex Functions
Neamtu, M., 1991, Enschede: University of Twente. 8 p. (Memorandum; no. 967)
Research output: Book/Report › Report › Professional

Cite this

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abstract = "Some negative results concerning the convexity preserving approximation and interpolation of multivariate functions are presented. We prove that the approximation based on both interpolation and local operators cannot be convexity preserving, provided the approximation space is (locally) finite dimensional. In both cases we can dispense with the asssumption of the linearity of the approximation operator and the assumption that the approximation space is a space of piecewise polynomials. Some consequences for the construction of shape preserving approximations are discussed.",

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AB - Some negative results concerning the convexity preserving approximation and interpolation of multivariate functions are presented. We prove that the approximation based on both interpolation and local operators cannot be convexity preserving, provided the approximation space is (locally) finite dimensional. In both cases we can dispense with the asssumption of the linearity of the approximation operator and the assumption that the approximation space is a space of piecewise polynomials. Some consequences for the construction of shape preserving approximations are discussed.

U2 - 10.1007/978-94-011-2634-2_28

DO - 10.1007/978-94-011-2634-2_28

M3 - Chapter

SN - 978-0-7923-1574-2

SN - 978-94-010-5164-4

T3 - Nato Science Series C

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BT - Approximation Theory, Spline Functions and Applications

A2 - Singh, S.P.

PB - Springer

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ER -

On Approximation and Interpolation of Convex Functions

Abstract

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Research output

On Approximation and interpolation of Convex Functions

Cite this