2012•PAMMRequires access

A note on the W1,p‐stability of piecewise linear interpolation

Thomas Dickopf

Abstract

Abstract It is known that piecewise linear interpolation of functions of one variable is uniformly bounded with an H1‐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 H1‐semi‐norm. In this note, a standard approximation argument is applied generalizing the result to piecewise linear interpolation of all functions in W1,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)

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Key concepts: Mathematics, Interpolation (computer graphics), Linear interpolation, Piecewise, Piecewise linear function, Counterexample, Bounded function, Norm (philosophy)