Some Ultrametric Caculus And Arithmetic On Local Fields

Following Amice,We know that there are Newton type bases{Q_n(x)}_{n∈N} on the dual space of the integer ring of the local field K,that f∈(O_K, K) ca,n be expanded as (?). P.E.Helsmoortel's caculation on the valuation of the differences of the polynomial functions Q_n(x),n∈N implies that if f(x)=(?) (x)∈C(O_K,K) and (?) = 0, then f is a C^j function on O_K Later, D.Barsky alsostudied the condition mentioned above for f to be a C^j-function, j∈N. But there might be problems on Barsky's paper. In this paper, We review some of Helsmoortel's caculation on the differences and Barsky's caculation on the Lipschitz condition, and give some estimates on the valuations of some related functions under certian concrete circumstances.In the last part of the paper, we discuss some interesting integral structures on the spaces of C^j function (j∈N) over local fields. We discuss the relation between the ring of formal power series and measure spaces,try to give some general discriptions about the measures, and some examples.