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A note on the W 1, p ‐stability of piecewise linear interpolation
Author(s) -
Dickopf Thomas
Publication year - 2012
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201210364
Subject(s) - mathematics , interpolation (computer graphics) , linear interpolation , piecewise , piecewise linear function , counterexample , bounded function , norm (philosophy) , constant (computer programming) , discrete mathematics , mathematical analysis , computer science , law , programming language , animation , computer graphics (images) , political science , polynomial
It is known that piecewise linear interpolation of functions of one variable is uniformly bounded with an H 1 ‐stability constant of one. In [1], we considered the nodal interpolation operator acting between spaces of piecewise linear functions and presented an elementary proof by minimizing a functional representing the H 1 ‐semi‐norm. In this note, a standard approximation argument is applied generalizing the result to piecewise linear interpolation of all functions in W 1,p on a real interval, 1 ≤ p ≤ ∞. We also comment on alternative proofs and finally give a counterexample for piecewise linear interpolation in 2D. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)