The Weierstrass-Stone approximation theorem for p-adic C^n-functions

Let K be a non-Archimedean valued field. T hen , on compact subsets of A", every Kvalued C n~function can be approxim ated in the C n-topology by polynomial functions (Theorem 1.4). This result is extended to a Weierstrass-Stone type theorem (Theorem 2.10). I N T R O D U C T I O N The non-archimedean version of the classical Weierstrass Approximation Theorem the case n = 0 of the Abstract. is well known and named after Kaplansky ([1], 5.28). To investigate the case ?? — 1 first let us re tu rn to the Archimedean case and consider a real-valued C 1-function ƒ on the unit interval. To find a polynomial function P such tha t b o th |ƒ —jP| and | / ' —P*\ are smaller or equal than a prescribed e > 0 one simply can apply the s tandard W eierstrass Theorem to ƒ' obtaining a polynomial function Q for which \ f f — Q\ < e. T h en re *-+ P ( . t ) := ƒ (0 ) + J0* Q(t)dt solves the problem. Now let ƒ : X —* K be a C 1-function where K is a non-archimedean valued field and X C K is compact. Lacking an indefinite integral the above m ethod no longer works. There do exist conti­ nuous linear antiderivations ([3], §04) b u t they do not m ap polynomials into polynomials ([3], Ex. 30.C). A further complicating factor is tha t the na tu ra l norm for C l -functions on X is given by ƒ ^ max {I/"(a*) | : . r 6 l ) V m a x i ^ ; x , y G X y x ^ y j I x —y J ra the r than the more' classical formula ƒ m a x { |/ ( . r ) : x 6 Ar ) V m a x { | / ' ( i ) | : x £ X } . (Observe th a t in the real case bo th formulas lead to the same norm thanks to the Mean Value Theorem , see [3], §§2G,27 for further discussions.)