The construction of normal bases for the space of continuous functions on Vq, with the aid of operators

Let a and q be two units of Zp, q not a root of unity, and let Vg be the closure of the set {aqn | n = o,1, 2, ...}. K is a non-archimedean valued field, K contains Qp, and K is complete for the valuation ) . ) , which extends the p-adic valuation. C(Vq --~ K) is the Banach space of continuous functions from Vq to K equipped with the supremum norm. Let E and Dq be the operators on C(Vq --> K) defined by = f(qx) and (Dqf)(x) = (f(qx) f (x))/(x(q-1 )). We will find all linear and continuous operators that commute with S (resp. with Dq), and we use these operators to find normal bases for --> K). If f is an element of C(Vq --. K), then there exist elements an o of K such that f(x) = 03A3 03B1nrn(x) where the series on the right-hand-side is uniformly n=o convergent. In some cases it is possible to give an expression for the coefficients . 1991 Mathematic3 subject classification : 46S10

= f(qx) and (Dqf)(x) = (f(qx) -f (x))/(x(q-1 )). We will find all linear and continuous operators that commute with S (resp. with Dq), and we use these operators to find normal bases for --> K). If f is an element of C(Vq --. K), then there exist elements an o of K such that f(x) = 03A3 03B1nrn(x) where the series on the right-hand-side is uniformly n=o convergent. In some cases it is possible to give an expression for the coefficients . 1991 Mathematic3 subject classification : 46S10 1. Introduction Let p be a prime, Zp the ring of the p-adic integers, Qp the field of the p-adic numbers.
K is a non-archimedean valued field, Qp, and we suppose that K is complete for the valuation ) . ), which extends the p-adic valuation. Let a and q be two units of Zp (i.e. |a| = == 1), q not a root of unity. Let Vq be the closure of the set = 0,1, 2, ...}. We denote by C(Vq --~ K) (resp. C(Zp --K) the set of all continuous functions f : Vq --> K (resp. f Zp --~ K) equipped with the supremum norm. If f is an element of C(Vq -~ K) then we define the operators E and Dq as follows : (£f )(x) = f(qx) We remark that the operator £ does not commute with Dq. . Furthermore, the operator Dq lowers the degree of a polynomial with one, whereas the operator E does not.
If ,C is a non-archimedean Banach space over a non-archimedean valued field L, and ei , e2,... is a finite or infinite sequence of elements of ~C, then we say that this sequence is orthogonal if ~~1e1 + ... + ~kek~ = max{~~iei~: 2 = 1,... ,&} for all k in N ( or for all k that do not exceed the length of the sequence ) and for all Ei, ... , fk in L. An orthogonal sequence el, e2, ... is called orthonormal if ~ei~ =1 for all i . A family (e=) of elements of ,C forms a(n) (ortho)normal basis of ,C if the family (e~) is orthonormal and also a basis . We will call a sequence of polynomials (pn(x)) a polynomial sequence if pn is exactly of degree n for all natural numbers n .
The aim here is to find normal bases for C(Vq ~ K), which consist of polynomial sequences. Therefore we will use linear , continuous operators which commute with Dq or with E. If is such a polynomial sequence , and if f is an element of C(Vq ~ K), there exist coefficients an in K such that f(x) = 03A3 03B1nrn(x) where the series on the right-n=0 hand-side is uniformly convergent. In some cases it is possible to give an expression for the coefficients an.
We remark that all the results (with proofs ) in this paper can be found in [5] , except for theorem 5 .

Notations.
Let Vq, K and C(Vq -~ K) be as in the introduction . The supremum norm on C(Vq -~i') will be denoted by ) ) . ) ) . We introduce the following : Ao(x) = 1, An(x) = (x -(n > 1), Bn(x) = = anqn(n-1)/2(q -1)nBn(x) It is clear that , (Bn(x)) and are polynomial sequences. The sequence (Cn(x)) forms a basis for C(Vq --~ K) and the sequence (Bn(x)) forms a normal basis for C(Vq ~ K). From this it follows that ~Bn~ = 1 and Let ~ and Dq be as in the introduction . Then we introduce the following : Definition. Let f be a function from Vq to K. We define the following operators : The operator Dq does not commute with D. The following properties are easily verified : is a polynomial of degree n , then (Dt~) p)(x) is a polynomial of degree n if n is at least j , and (D(j)p)(x) is the zero-polynomial if n is strictly smaller than j .
If f is an element of C(Vq --> h'), then we also have i) ( i) can be found in [1] , p. 60 , ii) can be found in [3] , p. 124-125 , iii) follows from i) and ii) ).

Linear Continuous Operators which Commute with £ or with Dq
Let us start this section with the following known result : If f is an element of C(Zp --; K), then the translation operator E on C(Zp --~ K) is the operator defined by Ef(x) = f ( x + 1) . .

Normal bases for
2014~ K) We use the operators of theorems 1 and 2 to make polynomials sequences (pn(x)) which form normal bases for 2014~ K). If Q is an operator as found in theorem 1 , with bo equal to zero, we associate a ( unique ) polynomial sequence with Q. We remark 00 that the operator R = ~ does not necessarily lowers the degree of a polynomial. Remark. Here we have |cn| in contrast with theorem 3, where we need |bn| |bN| (n > N).
If ~~ is an operator as found in theorem 4 , with N equal to one, then we can prove a theorem analogous to theorem 2 : Theorem 5 Let Q be an operator such that Q = ~ciDiq, with |c1 ( = -1)|, i=1 |cn| ~ | ( q -1 ) n | if n > 1, , and let pn(x) be the polynomial sequence as found in theorem 4. An operator T on C(Vq -K) is continuous , linear and commutes with Dq if and only ilT is of the form T = ~diQi , where the sequence (dn) is bounded, where dn = (Tpn)( a). i=o Remark.
In theorem 2 the sequence {cn~{q -1)") must be bounded, whereas here the sequence (dn) must be bounded. This follows from the fact that the norm of the operator Dq equals q -1 (' 1 , whereas the norm of the operator Q equals 1 . ..

More Normal Bases
We want to make more normal bases, using the ones we found in theorems 3 and 4 . For operators which commute with E we can prove the following theorem : Theorem 6 Let be a polynomial s equence which forms a normal basis for C(Vqk'), and let Q = ~biD(i) (N > 0) with 1= |bN| > |bk| if k > N. If Qpn(x) = xNrn-N(x) i=N (n > N), then the polynomial sequence (rk(x)) forms a normal basis for C(Vq -K) .
And analogous for operators which commute with the operator Dq we have : T heore m 7 Let {pn(x)) be a polynomial s equence which forms a normal basis for C(Vq ~ K), 00 and let Q = ~ciDiq (N > 0) with |cN| = I(q , l if n > N. i~N If = rn..N(x) (n > N), then the polynomial sequence (rk(x)) forms a normal basis for C(Vq -K) .
We remark that analogous results can be found on the space C(Zp -K) for linear continuous operators which commute with the translation operator E. The result analogous to theorems 3 and 4 for the case N equal to one, was found by L. Van Hamme (see (4~), and the extensive version of theorems 3 and 4, and the analogons of theorems 5, 6 and 7 can be found with proofs similar to the proofs of the theorems in this paper.