The Weierstra55-stone Approximation Theorem for P-adic Cn-functions

Let K be a non-Archimedean valued field. Then, on compact subsets of A", every K-valued C n~function can be approximated in the Cn-topology by polynomial functions (Theorem 1.4). This result is extended to a Weierstrass-Stone type theorem (Theorem 2.10).

The non-archimedean version of the classical Weierstrass Approximation Theorem -the case n =0 of the Abstractis well known and named after Kaplansky ([!}, 5.28). To Investigate the case ~ == 1 first let us return to the Archimedean case and consider a real-valued C1-function f on the unit interval. To find a polynomial function P such that both and are smaller or equal than a prescribed ~ > 0 one simply can apply the standard Weierstrass Theorem to /' obtaining a polynomial function Q for which Then x P(x) := Q(t)dt solves the problem. Now let f X a C1-function where K is a non-archimedean valued field and X c K is compact.
Lacking an indefinite integral the above method no longer works. There do exist continuous linear antiderivations ([3j, §64) but they do not map polynomials into polynomials ([3], Ex. 30.C). A further complicating factor is that the natural norm for C1-functions on X is given by . rather than the more classical formula (Observe that in the real case both formulas lead to the same norm thanks to the Mean Value Theorem, see [3], § §26,27 for further discussions.) , Thus, to obtain non-archimedean Cn-Weierstrass-Stone Theorems for n E (1, 2, ...~ our methods will necessarily deviate from the 'classical' ones.. 0. PRELIMINARIES , 1. Throughout K is a non-archimedean complete valued field whose valuation J ) is not trivial. For a E K, r > 0 we write B(a, r~ := ~~ E K : r}, the 'closed' ball about a with radius r. 'Clopen' is an abbreviation for 'dosed and open'. The function x ~ ~ (z E K) is denoted X. The K-valued characteristic function of a subset Y of K is written ~Y. For a set Z, a function f : Z --~ K and a set W C Z we define z e W~ (allowing the value ). The cardinality of a set F is No := ~~,1,2,...~, N := (1,2,3,...~. We now recall some facts from ~~j, ~3~ on C"-theory.. , 2. For a set Y' C K, n E N we set := ((~, ~t2, . .. , yn~ y Yn : z~j ===~ ys~~~~K , n E No we define its nth difference quotient  (iv) If f e C'*(X -~ K) then for .c, y X we have Taylor's formula where 4. Since JC is compact the deference quotients (0 t ~ n) are bounded if f C"(X -~ K). We set Then = We quote the following from ~2j and ~31. Recall that a function f : X --r is a local polynomial if for every a E-X there is a neighbourhood U of a such that f ~ X fl U is a polynomial function. In the sequel we need an extension of this formula to finite products of functions. The proof is straightforward by induction with respect to N. The following key lemma grew out of ~~~, 5.28. > 1, so we only have to deal with j, = 0 (then j.X L X) or j, = 1 (then = I }. The latter case occurs j times (as ~ j, = j) and it follows that ==1 k JY (~ j~ X }(zs) is a product of k-j distinct terms taken from {xi, ... , (observe s=1 that, indeed, j k since j n k), so its absolute value It follows that from which we conclude . a e and step 1 is proved.
Step that is a sum of terms of the form where j1 + ~ ~ ~ + j, = j . If = 0 it follows from Step 1 that the absolute value of (5) is _ ~ 6-j.( ïtJ)i where the product is taken over all s in the nonempty set r ,:= {s E ~1, ... , t~} : j~ > 0}, so the product is _ If > 0 it follows from Step 1 that the absolute value of (5) is t~ ~'j. ~I ~ w . follows from Step I that the absolute value of ~) is II 6~" ~ = 6~' : " . The statement of Step 2 follows.
Step 3. Proof of (4). Again, the Product Rule 1.1, now applied to h~ =    Remark. It follows directly that the local polynomial functions X --; K form a dense subset of --> h'). Remarks.
1~. The case n = 0 yields, at least for those X that are embeddable into ~.', the well known Kaplansky Theorem proved in [1], 6.15.
2. yVe leave it to the reader to establish a C°°-version of Theorem 2.10.