(untitled) (Analysis in Theory and Applications, 2017)
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(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^σ(K_p), σ>0, are proved; then the equivalent approximation theorem on Sobolev-type space W^r_σ(K_p), σ≥ 0, 1≤ r