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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A measure of smoothness related to the Laplacian
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by Z. Ditzian PDF
Trans. Amer. Math. Soc. 326 (1991), 407-422 Request permission

Abstract:

A $K$-functional on $f \in C ({R^d})$ given by \[ \tilde K (f,{t^2})= \inf (||f - g|| + {t^2}||\Delta g||;g \in {C^2} ({R^d}))\] will be shown to be equivalent to the modulus of smoothness \[ \tilde w (f,t)= \sup \limits _{0 < h \leq t} \left \| {2 df(x) - \sum \limits _{i = 1}^d {[f(x + h{e_i}) + f(x - h{e_i})]} } \right \|.\] The situation for other Banach spaces of functions on ${R^d}$ will also be resolved.
References
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 326 (1991), 407-422
  • MSC: Primary 41A25
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1068926-5
  • MathSciNet review: 1068926