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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A measure of smoothness related to the Laplacian


Author: Z. Ditzian
Journal: Trans. Amer. Math. Soc. 326 (1991), 407-422
MSC: Primary 41A25
MathSciNet review: 1068926
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Abstract: A $ K$-functional on $ f \in C\,({R^d})$ given by

$\displaystyle \tilde K\,(f,{t^2})= \inf (\vert\vert f - g\vert\vert + {t^2}\vert\vert\Delta g\vert\vert;g \in {C^2}\,({R^d}))$

will be shown to be equivalent to the modulus of smoothness

$\displaystyle \tilde w\,(f,t)= \mathop {\sup }\limits_{0 < h \leq t} \,\left\Ve... ...,df(x) - \sum\limits_{i = 1}^d {[f(x + h{e_i}) + f(x - h{e_i})]} } \right\Vert.$

The situation for other Banach spaces of functions on $ {R^d}$ will also be resolved.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1068926-5
PII: S 0002-9947(1991)1068926-5
Keywords: Laplacian, $ K$-functional, smoothness
Article copyright: © Copyright 1991 American Mathematical Society