The Smoothness of Fractal Interpolation Functions onℝand onp-Series Local Fields

Abstract

A fractal interpolation function on ap-series local fieldKpis defined, and itsp-type smoothness is shown by virtue of the equivalent relationship between the Hölder type spaceCσKpand the Lipschitz class Lipσ,Kp. The orders of thep-type derivatives and the fractal dimensions of the graphs of Weierstrass type function on local fields are given as an example. Theα-fractal function onℝis 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 onℝandKpis given.