Locally Constant Functions on Zp 1. Review of General Properties1.Review of General Properties

2.AEF,A'DA=A'EY

A.4. Circular Filters

3. Locally Constant Functions on Zp 1. Review of General Properties1.Review of General Properties

coo

chi

`ti

Take now x _ -m in this equality:

Sf(0) - Sf(-m) = f(-m)+... + f(-I)

= f(a(m - 1))+...+f(a(0))

= S(f o a)(m ).

Since Sf (0) = 0, the result follows.

Example. Let a = 1+ t E l+ M C Cp and take

f(x)=ax =(I+t)x=Y,

k>O

Then we have

x k

Sf(x)

tk(k +

(1 +ttx - 1 _{(t 54 0),}

k>0

ax- I

Sf(x)= a-1 ^(a01).

3. Locally Constant Functions on Zp

A

A

3. Locally Constant Functions on Zp 179

pxooF. Give E the discrete metric. Since X is compact, f : X -> E is uniformly continuous. Hence there is a S > 0 such that

d(x, y) < S = d(f (x), f (y)) < 1 f (x) = f (y), and the conclusion follows.

The set of functions X -* E is denoted by J'(X; E), and when E = K is a field, .1(X; K) = J7(X) is a vector space over K (omitted from the notation if this field is implicit from the context). When X is a compact ultrametric space, the locally constant functions X -> K form a K-vector subspace .1'c(X) of .1(X).

The characteristic functions of clopen balls of X form a system of generators of

Pcm-

3.2.

Characteristic Functions of Balls of Zp

We are interested in locally constant functions on X = Zp taking values in any abelian group M (this abelian group will typically be an extension K of Qp). Let us start by the study of the (uniformly) locally constant functions f E .Ik(Z p; M)

satisfying

Ix-YI <IP'I=Pj

I¹⁷

' f(x)=f(y)

for some fixed integer j > 0. These are the functions that are constant on all closed balls of radiusr j = 1/pd. Since the balls in question are the cosets of pj Z P in Zp, these functions are the elements of the vector space

Fj = -7:7(Z/p3Z) = J"(Zp/P'Zp) C .1t°(Zp; K).

In fact,we have a partition

Zp=

1_L

(i+p3Zp)

O<i <p,

into balls of radiusrj, and

i + p-'Zp = B<p-;(i)

(0 < i < pi)

is anenumeration of these balls in Zp. For fixed j the characteristic functions lpi+j = characteristic function of the ball B<1 p, (i) (0 < i < pj )

makeup a basis of the finite-dimensional space Fj. When we let j increase, the subspacesFj also increase, and

Fk(Zp; K) = U Fj-

j>O

Unfortunately,the previously given basis of Fj has no element in common with the basisconstructed similarly in Fj_1. A clever way of constructing coherent bases

.may OOHOOHOOH

"i3 ...

of the spaces Fj - where the basis of Fj extends the basis of Fj_t - has been devised by M. van der Put. Let

cPo.o = 1 characteristic function of Z_P (i = 0)

1/ri = _coil characteristic function of i + pZp

(1 < i < p)

cp;,2 characteristic function of i + p2Zp

(p < i < p2),

etc.

Generally,

1lri = cpi, j characteristic function of i + pJZp if pj-1 < i < pJ.

Since absolute values of elements of Zp can only be powers of p, we have

Ixl<

¹

Ixl< lj

i

p

^(Pf

-r < i < pi),

and i/ii = cpi, j is also the characteristic function of the ball

Bi={xEZp:Ix-iJ <1/i)

(with the convention B0 = Z_Pfor i = 0).

On the other hand, the indices i in the range pJ-1 < i < pi are precisely those that admit an expansion of length j in base p, namely an expansion of the form

i=io+i1p+...+ij-1Pt-1

(0 ie

P-1,ij-1o0).

Definition. The length of an integer i > 1 is the integer v = v(i) > 1 such that the expansion of i in base p has digits ie = O for £ > v, while iv-1 ¢ 0.

With this definition, the van der Put sequence is defined by

*j = Wi,v(i) : characteristic function of i + p'(i)Zp Here are the first few functions:

1

V0,r

...

_cpp-1,1

VO,2

...

_epp-1,2 Vp,2

...

4'p2-1 2

c°o,3

...

_Vp-1,3 ^...

... ...

_(Pp2,3

...

Wp3-1,3

cp0,j

The sequence (* )i<p; appears at the top of this triangular table of characteristic functions.

Proposition. The sequence (*j )o<i <p, is a basis of F1 (j > 0).

3. Locally Constant Functions on Zp 181

PROOF For fixed j > 0,the components of any f E Fj in the known basis ii,j (i < pj) of Fj are the (constant) values off on the balls i + piZp:

f = > f(i)cpi,j-

i<p,

In particular, for f = Ire, the characteristic function of Be = .£ + p°Zp, we have a sum of the form

where the indices that occur are the same as those occurring in the partition

BE = U (i + p3Zp)

They are the indices i such that 0 < i < pi and i - .£ (mod p)" (in order to have i E Be). These indices can be listed:

i =2, £+p", £+2p", ....

The first one is f itself, and they are all greater than or equal to £. The matrix of the components of the 1,Ire in the basis Vi,j is lower triangular with l's on its diagonal (all its entries are 0's and l's). This matrix U has determinant 1 and hence is invertible: The *t (0 < £ < pi) form a basis of Fj. If we write U

= I + N,

the matrix N is lower triangular with 0's on its diagonal and hence is nilpotent: A power of N vanishes. This proves that

U-t = I - N +

^N2

_ ... + (-1)mNm if

Nm+t = 0.

In particular, the inverse U-1 of U has integral entries: The components of the'p,, j in the basis (ilre)are also integers.

Here is aneven more precise result.

Proposition. If f = E ai i,ri E Fj, the coefficients are given by ao = f(0) and a» = f(n) - f(n-) (n > 1),

where n_

= n -

pv-' denotes the integer of length strictly smaller than n obtained by deleting itstop digit in base p.

PROOF We have already observed that f (0) = ao. Fix a positive integer n and

considerthe sum f(n) _ >i<p, a i/it (n) in which di(n) = 0 or 1. More precisely,

i/ri(n)=1nEBi

min - i mod p"(`)

the digits of n and i are the same up to v(i)

t i is an initial partial sum of n.

This shows that

f (n) = ao + (*) + an, whereas

f(n-) = ao + (*)- Hence f(n) - f(n_) = an as claimed in the proposition.

Corollary. When f = ai>Jri E Fj takes its values in an ultrametric field, we have

Il f 11 =max Jail .

PROOF For each x E ZP we have *j (x) = 0 or 1: 1*i(x)I < 1 and If (x) J = I E ai ii (x)I << max Jai 1.

This proves

Il f IJ = sup If (x) J < max Jai 1.

Conversely, ao = f (0) = Iaol

11f II, and for n > 1,

Ianl = I f(n) - f(n-)I < max(If(n)I, If(n-)U < IJf1I, hence maxIan I < IIfJI-

Since (iJri)i>o is a basis of Y"(Zp, K) = Uj>o Fr, it is easy to generalize the preceding results to all locally constant functions (taking their values in an extension K of Qp).

Theorem. Let f : ZP -+ K be a locally constant function. Define

ao = f(0), a n = f(n) - f(n-) (n> 1).

Then f =

aiifi is a finite sum and 11 f II = supi Iai I.

3.3. The van der Put Theorem

We are now able to give the main result, namely the representation of any contin- uous f : ZP -* K where K is a complete extension of Qp.

Theorem. Let f : ZP -+ K be a continuous function. Define 40 = f (0), an = f (n) - f (n-) (n > I)-

88w

4. Ultrametnc Banach Spaces 183 Then la, I 0, and > ai Ij converges uniformly to f. Moreover,

If II = sup Jai I = max Jai I.

i i

PROOF Since In - n_ 1 0 (n -* oo) and f is uniformly continuous, we have Ian I = If(n) - f (n _ )I - 0 (n -* oo), and the series converges uniformly. The sum of this series is a continuous function,

g=Eai'`i.

We still have to prove f = g. Since these functions are continuous, it is enough to show that their restrictions to the dense subset N are the same. The obvious equality f (0) = ao = g(0) can be used as the first step in an induction on n. Let 91 = Fi<p; aiY'i. Forn < pi we have

f(n) - f(n_) = a

(by definition)

= coefficient of gj (since n < pi)

= gi(n) - gi(n-), f(n) - gi(n) _ .f(n-) - gi(n-)-

This shows that if f and g, agree on (0, 1, 2,

... , n - 11,

they will also agree at the point n (provided that n < pi ). As a consequence, for all integers n E N, f(n) = limb gi(n) = g(n) (with a stationary convergence). As mentioned, this proves f = g. The equality Il f II = supi Jai Iis obtained exactly as in the case f locally constant.