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Mahler Expansions MAHLER EXPANSIONS KEITH CONRAD 1. Introduction The set C([0; 1]; R) of all continuous functions f : [0; 1] ! R is a basic example of a function space in real analysis, and its p-adic analogue is the set C(Zp; Qp) of all continuous functions f : Zp ! Qp. These are analogues both because [0; 1] and Zp are compact and because R and Qp are both complete. Examples of continuous functions are polynomials on [0; 1] with real coefficients and polynomials on Zp with p-adic coefficients. A metric can be put on C([0; 1]; R) and C(Zp; Qp) by measuring the distance between two functions as the largest distance between their values: 8 < max jf(x) − g(x)j if f; g : [0; 1] ! R; (1.1) d(f; g) = x2[0;1] :max jf(x) − g(x)jp if f; g : Zp ! Qp: x2Zp The maximum really occurs because jf(x) − g(x)j in the real case and jf(x) − g(x)jp in the p-adic case are continuous functions [0; 1] ! R and Zp ! R, and any real-valued continuous function on a compact metric space has a maximum value (Extreme Value Theorem). Theorem 1.1. The function d in (1:1) is a metric and the spaces C([0; 1]; R) and C(Zp; Qp) are complete with respect to it. Proof. See the appendix. In both C([0; 1]; R) and C(Zp; Qp), the polynomial functions are a dense subset: (1) if f : [0; 1] ! R is continuous and " > 0 there is a polynomial p(x) with real coefficients such that jf(x) − p(x)j < " for all x 2 [0; 1]. (2) if f : Zp ! Qp is continuous and " > 0 there is a polynomial p(x) with coefficients in Qp such that jf(x) − p(x)j < " for all x 2 Zp. In the real case, the denseness of polynomials in C([0; 1]; R) is due to Weierstrass1 (1885). The denseness of polynomials in C(Zp; Qp) was first proved by Dieudonne (1944). While there is no nice description of all the functions in C([0; 1]; R) (they are not describable as power series, since power series are infinitely differentiable and a continuous function need not be even once differentiable everywhere), in 1958 Mahler [3] gave a very nice description of all functions in C(Zp; Qp) using infinite series of special polynomials. Theorem 1.2 (Mahler). Every continuous function f : Zp ! Qp can be written in the form X x x x (1.2) f(x) = a = a + a x + a + a + ··· n n 0 1 2 2 3 3 n≥0 1This was later generalized by Stone to a result now called the Stone{Weierstrass theorem, describing conditions under which a suitable collection of continuous real or complex-valued functions on a compact space is dense in the set of all continuous functions on that space. 1 2 KEITH CONRAD for all x 2 Zp, where an 2 Qp and an ! 0 as n ! 1. We'll see there is a formula for the coefficients an in terms of values of f that is reminiscent of Taylor's formula for the coefficients of a power series, but it must be stressed that (1.2) represents an arbitrary continuous function, not a function expressible by a power series. x Just because each n is differentiable (even infinitely differentiable) does not mean the series in (1.2) has that property too. This is similar to the \paradox" of Fourier series, where continuous functions [0; 1] ! R have a representation as an infinite series of sines and cosines, but need not be differentiable even though sin(nx) and cos(nx) are differentiable. The expansion (1.2) is called the Mahler expansion of f and the numbers an are called PN x the Mahler coefficients of f. If fN (x) is the polynomial n=0 an n then jf(x) − fN (x)jp = P x j n≥N+1 an n jp ≤ maxn≥N+1 janjp since binomial coefficients at p-adic integers are p-adic integers. This maximum goes to 0 as N ! 1 since an ! 0, so Mahler's theorem is an explicit (and useful!) way of approximating continuous functions Zp ! Qp by polynomials. 2. Continuity and coefficient formula Before we show every continuous function Zp ! Qp has a Mahler expansion, let's see P x why an infinite series n≥0 an n with an ! 0 in Qp is a continuous function on Zp. P x Step 1: When an ! 0 in Qp, the infinite series n≥0 an n converges for all x 2 Zp. x The key point is that even though n = x(x − 1) ··· (x − (n − 1))=n! has n! in the denominator, which would ordinarily be bad p-adically (because n! is very small p-adically x as n grows), it is not bad when x 2 Zp because n 2 Zp by p-adic continuity of polynomials x x and its values on the dense subset N. Thus j n jp ≤ 1 for x 2 Zp, so jan n jp ≤ janjp, which x P x proves jan n jp ! 0 from janjp ! 0. Therefore n≥0 an n converges in Qp for each x 2 Zp. P x Step 2: When an ! 0 in Qp, the function f : Zp ! Qp defined by f(x) = n≥0 an n is continuous. For x0 2 Zp we want to prove f is continuous at x0. Pick " > 0. Since an ! 0, there is an x N such that janjp < " for n ≥ N. Each of the finitely many functions n for 0 ≤ n ≤ N − 1 is continuous at x0, so by picking the minimal δ used for each of them in the "-δ definition of continuity at x0, there is a single δ > 0 such that x x0 (2.1) jx − x0jp < δ =) − < " n n p for n 2 f0; 1;:::;N − 1g. If jx − x0jp < δ, X x x0 x x0 jf(x) − f(x0)jp = an − ≤ max janjp − : n n n≥0 n n n≥0 p p x x0 x x0 Since n − n 2 Zp we have janjpj n − n jp ≤ janjp < " for n ≥ N. For the terms x x0 preceding the Nth term, we instead say janjp n − n p ≤ janjp" by (2.1). Let A = maxn≥0 janjp (this exists since the an's tend to 0), so janjp ≤ A for all n. Thus jx − x0jp < δ =) jf(x) − f(x0)jp ≤ max("; A") = max(1;A)"; so f is continuous at x0. As x0 was arbitrary, f is continuous on Zp. MAHLER EXPANSIONS 3 P x Next we will derive a formula for the coefficients in f(x) = n≥0 an n if an ! 0 in Qp. It is easy to get the value of the constant term: setting x = 0 in (1.2) we obtain a0 = f(0) since all terms vanish in the Mahler expansion except the one at n = 0. Similarly, if x = 1 then all terms in (1.2) vanish except for the first two: f(1) = a0 +a1, so a1 = f(1)−a0 = f(1)−f(0). Setting x = 2, all terms in (1.2) vanish except the first three: f(2) = a0 + 2a1 + a2, so a2 = f(2) − a0 − 2a1 = f(2) − f(0) − 2(f(1) − f(0)) = f(2) − 2f(1) + f(0): These formulas suggest we can write an in terms of f(0); f(1); : : : ; f(n), and we will see this is true. The basic mechanism behind a formula for an is the discrete difference operator ∆: for any function f : Zp ! Qp, define ∆f : Zp ! Qp by (∆f)(x) = f(x + 1) − f(x): n This is also a function Zp ! Qp, and it can be iterated to have functions ∆ f for n ≥ 2: ∆2f = ∆(∆f), and more generally ∆nf = ∆(∆n−1f). Set ∆0f = f, which is analogous to the zeroth derivative f (0) of a function f being the function itself. The discrete difference operator ∆ behaves nicely on the binomial coefficient polynomials x x x because it shifts them down by one: ∆ n = n−1 for n ≥ 1, and ∆ 0 = ∆(1) is the zero function. Indeed, by the Pascal's triangle recursion for binomial coefficients, if n ≥ 1 then x x + 1 x x x x x ∆ = − = + − = : n n n n − 1 n n n − 1 This resembles the effect of differentiation on xn:(xn)0 = nxn−1 for n ≥ 1, and (xn)0 = 0 for n = 0. (A more accurate analogy would be with differentiation on the functions xn=n!, n−1 whose derivative is x =(n − 1)! for n ≥ 1.) Applying ∆ to a function f : Zp ! Qp having P x a Mahler expansion n≥0 an n where an ! 0, X x + 1 X x X x + 1 x X x (∆f)(x) = a − a = a − = a ; n n n n n n n n n − 1 n≥0 n≥0 n≥0 n≥1 where a0 drops out and other coefficients shift down one position. Reindexing the series to start at n = 0, X x X x x x ∆ a = a = a + a x + a + a + ··· : n n n+1 n 1 2 3 2 4 3 n≥0 n≥0 The effect of applying ∆ to a Mahler expansion m times is to shift coefficients m positions: X x X x x x ∆m a = a = a + a x + a + a + ··· : n n n+m n m m+1 m+2 2 m+3 3 n≥0 n≥0 Setting x = 0 leaves only the constant term am, so we have proved the following theorem.
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