Construction Theory of Function on Local Fields

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.