Year: 2015
Analysis in Theory and Applications, Vol. 31 (2015), Iss. 1 : pp. 25–44
Abstract
We establish the construction theory of function based upon a local field $K_p$ as underlying space. By virtue of the concept of pseudo-differential operator, we introduce "fractal calculus" (or, $p$-type calculus, or, Gibbs-Butzer calculus). Then, show the Jackson direct approximation theorems, Bernstein inverse approximation theorems and the equivalent approximation theorems for compact group $D(\subset K_p)$ and locally compact group $K^+_p(=K_p)$, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the Hölder-type space $C^\sigma(K_p), $ $\sigma>0$, are proved; then the equivalent approximation theorem on Sobolev-type space $W^r_\sigma(K_p),$ $ \sigma\geq 0,$ $ 1\leq r<+\infty$, is shown.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2015.v31.n1.3
Analysis in Theory and Applications, Vol. 31 (2015), Iss. 1 : pp. 25–44
Published online: 2015-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Construction theory of function local field fractal calculus approximation theorem Hölder-type space.
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