| Let p be a rational prime number, and let r be a power of p. Let ℘ be a prime ideal in the polynomial ring Fr T over the finite field Fr . Suppose ℘ is generated by a monic polynomial p(T) of degree d, then we have an absolute ˙ on k = Fr T which is normalized by &vbm0;p&parl0;T&parr0;&vbm0;=1/q with q = rd. The valuation corresponding to this absolute value is denoted v, and v(p(T)) = 1 by the normalization. Denote O=Fr T℘ as the completion of Fr T at ℘ , and L = Fr T ℘ as the completion of Fr T at ℘ . Let Cm&parl0;O,L&parr0; denote the space of Cm-functions from O to L which is defined by Schikhof in [W. Schikhof, Ultra-metric calculus: an introduction to p-adic analysis. Cambridge Stud. Adv. Math., Vol. 4, Cambridge Univ. Press, Cambridge, UK, 1984], and let LA( O , L) denote the space of locally analytic functions on O with values in L. Also denote {lcub}Gn( x){rcub}n ≥ 0 the sequence of Carlitz polynomials (defined as in Section 8.22 of [D. Goss, Basic structures of function field arithmetic, Springer-Verlag Berlin Heidelberg 1998]), which is an orthonormal basis of the space of continuous functions C( O , L) = C0( O , L). We prove that if f&parl0;x&parr0;=n =0infinityan Gn&parl0;x&parr0;∈C&parl0; O,L&parr0; , then (i). f&parl0;x&parr0;∈Cm&parl0;O ,L&parr0; if and only if limn→infinity an nm=0. (ii). f&parl0;x&parr0;∈LA&parl0;O, L&parr0; if and only if liminfn v an n >0.; From this result, we describe the dual spaces of Cm&parl0;O,L&parr0; and LA( O , L) in terms of divided power series, and in terms of the dual spaces of locally polynomial functions (the space of distributions is the dual space of locally constant functions). Finally, we study the measures which yield the Goss zeta function on Fr T via integrals of power functions. |
