The Smoothness of Fractal Interpolation Functions on R and on p-Series Local Fields

A fractal interpolation function on a p-series local field K-p is defined, and its p-type smoothness is shown by virtue of the equivalent relationship between the Holder type space C-sigma (K-p) and the Lipschitz class Lip(sigma, K-p). The orders of the p-type derivatives and the fractal dimensions of the graphs of Weierstrass type function on local fields are given as an example. The alpha-fractal function on R is introduced and the conclusion of its smoothness is improved in a more general case; some examples are shown to support the conclusion. Finally, a comparison between the fractal interpolation functions defined on R and K-p. is given.