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

A fractal interpolation function on a p-series local field Kp is defined, and its p-type smoothness is shown by virtue of the equivalent relationship between the Hölder type space CσKp and the Lipschitz class Lipσ,Kp. 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 α-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 Kp is given