A measure of smoothness related to the Laplacian

Ditzian, Z.

doi:10.1090/S0002-9947-1991-1068926-5

A measure of smoothness related to the Laplacian
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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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