TY - JOUR
T1 - Construction Theory of Function on Local Fields
JO - Analysis in Theory and Applications
VL - 1
SP - 25
EP - 44
PY - 2017
DA - 2017/01
SN - 31
DO - http://doi.org/10.4208/ata.2015.v31.n1.3
UR - https://global-sci.org/intro/article_detail/ata/4620.html
KW - Construction theory of function, local field, fractal calculus, approximation theorem, Hölder-type space.
AB - <p style="text-align: justify;">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 &quot;fractal calculus&quot; (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), $ $\sigma&gt;0$, are proved; then the equivalent approximation theorem on Sobolev-type space $W^r_\sigma(K_p),$ $ \sigma\geq 0,$ $ 1\leq r&lt;+\infty$, is shown.</p>