Let $K$ be a non-archimedean valued field which contains $\Bbb Q_p$ and suppose that $K$ is complete for the valuation $|\cdot|$, which extends the $p$-adic valuation. $V_q$ is the closure of the set $\{aq^n|n=0,1,2,\dots\}$ where $a$ and $q$ are two units of $\Bbb Z_p$, $q$ not a root of unity. $C(V_q\rightarrow K)$ is the Banach space of continuous functions from $V_q$ to $K$, equipped with the supremum norm. Our aim is to find normal bases $(r_n(x))$ for $C(V_q\rightarrow K)$, where $r_n(x)$ does not have to be a polynomial

Verdoodt, A.

Revistes Catalanes amb Accés Obert

Publicacions Matemátiques, Vol 38 (1994), 371-380 .
NORMAL BASES FOR THE SPACE
OF CONTINUOUS FUNCTIONS
DEFINED ON A SUBSET OF zp
ANN VERDOODT
A bstract
Let K be a non-archimedean valued field which contains Qp and
suppose that K is complete for the valuation • 1, which extends the
p-adic valuation . Vq is the closure of the set {agn in = 0, 1, 2, . . . }
where a and q are two units of Zp, q not a root of unity. C(Vq -->
K) is the Banach space of continuous functions from Vq to K ,
equipped with the supremum norm . Our aim is to find norma l
bases (rn (x)} for C(Vq --~ K), where rn (x) does not have to be a
polynomial .
1 . Introduction
The main aim of this paper is to find normal bases (r(x)) for the
space of continuous functions on Vq , where rn (x) does not have to be a
polynomial .
Therefore we start by recalling some definitions and some previous
results .
Let E be a non-archimedean Banach space over a non-archimedean
valued field L .
Let f1 , f2 , . . . be a finite or infinite sequence of elements of E. We say
that this sequence is orthogonal if IIaifi + • • • +

fk II = max{ i i aczfi l j
i = 1, . . . , k} for all k in N (or for all k that do not exceed the length of
the sequence) and for all a l , . . . , ak in L . If the sequence is infinite, it
follows that E ai fi = max{iia ifi : i = 1, 2, . . . } for all a l , a2, . . . in
i= z
L for which
i
hm aifi = O . An orthogonal sequence f1 , f2 , . . . is called
orthonormal if 11 fi II = 1 for all ~ .
This leads us to the following definition :
372

A . VERDOOD T
If E is a non-archimedean Banach space over a non-archimedean val-
ued field L, then a family (fi) of elements of E is a (ortho)normal basi s
of E if the family (fi) is orthonormal and also a basis .
An equivalent formulation is (see [1, Propositions 50 .4 and 50 .61 )
If E is a non-archimedean Banach space over a non-archimedean val-
ued fieid L, then a family (fi) of elements of E is a (ortho)normal basi s
of E if each element x of E has a unique representation x = Exifi where
xi E L and xi --> 0 if i oo, and if the norm of x is the supremum of
the norms of xi .
Let 7Lp be the ring of p -adic integers, Qp the fieid of p-adic numbers ,
and K is a non-archimedean valued fieid, K containing Qp, and we sup-
pose that K is complete for the valuation ~ • ~ , which extends the p-adic
valuation . Let a and q be two units of Zp , q not a root of unity. We define
Vq to be the closure of the set {aqnln = O, 1, 2, . . . } . The set Vq has been
described in [3] . Let C(Vq - -} K) (resp . C(Lp ---~ K)) be the Banach
space of continous functions from Vg to K (resp. Zp to K) equipped
with the supremum norm . N denotes the set of natural numbers, and No
is the set of natural numbers without zero .
We introduce the following :
If x is an element of Qp , x can be written in the following way :
+oo
x = E aipj where is zero for i sufficiently large (i E N) (see
i_—oc
[1, section 3 and section 4]) . This is called the Henseldevelopment of
the p-adic integer x . We then define the p-adic entire part [x] 1, of x by
1 n- 1
[x] p = E aipa and we put xn — pn [p—nx]p — E ajpi (n E N) .
.7=-00 i=-oo
We write m q x, if m is one of the numbers xo, x 1 , . . . . We then say
that "m is an initial part of x" or "x starts with m" (see [1, section 62D..
s
If n belongs to No, n = E aip2 where as 0, then we put n_ =
j=o
s- 1
E cl ip.) . We remark that n_ q n .
j=o
In [1, Theorem 62 .2], we find the following result which is due to van
der Put :
Theorem.
The functions go, gla . . . defined by
gn(x) =1 ifn q x ,
= 0 otherwise,
NORMAL BASES

373
foren a normal basis for C(7L P —> K) . If f is an element of C(7G P —> K) ,
00
then f can be written as a uniformly convergent series f (x) = E rykgk (x )
k= 0
where 'yo = f (0) and ryn = f (n) — f (n_) if n E Np .
We now survey the content of this paper :
In Theorem 1 of section 2, our aim is to find a basis (e(x)) analogous
to van der Put's basis, but with the space C (7Lp --- + K) replaced by
C(Vq —} K) . If f is an element of C(Vq ~ K), then there exist elements
ak of K such that f (x) = Eakek(x) where the series on the right-hand-
k=o
side is uniformly convergent . We are able to give an expression for the
coefficients a k .
In Theorem 2 of section 3, we prove the following result :
n
Define rn (x) = E cn ;jej (x), cn ;j E K, Cn ;n 0 ((e(x)) as in Theo-
j=o
rem 1 below) .
Then (r(x)) forms a normal basis for C(Vq ~ K) if and only if fo r
all n f ~rnll = 1 and Icn ;n1 = 1 .
In Theorem 3 of section 3, we give an extension of Theorem 2 :
Let (rn (x)) be such a sequence which forms a normal basis fo r
n
C(Vq ---} K), and let (sn(x)) be a sequence such that sn (x) = E d n;jrj (x) ,
j=o
dn ; j E K, dn;n O . Then (s(x)) forms a normal basis for C( Vg --} K) <=>
II Snll = 1, k1n ;n I = 1 <=> Idn ;j 1 ç 1, Idn ;n I = 1 .
Acknowledgement . I thank professor Van Hamme for the advice he
gave me during the preparation of this paper .
2 . Proof of the first theorem
We start with some lemmas and some definitions.
Definition .
If b and x are elements of Z, b 1 (mod p), then we put b~ = lim b n .
n--} x
The :mapping: 7Lp ---} 7L p : x ~ bx is continuous.
For more details, we refer the reader to [1, section 32] .
Not ation .
Take m ~ 1, m the smallest integer such that qm 1 (rnodp) .
We have 1 ç m ç p -- 1 and (qm)X is defined for all x in 7L .
374

A. VERDOOD T
Definition .
Let k be a natural number prime to p . We denote by 7LP (k) the pro-
jective limit 7Lp (k) = lim(7L/kpj7L) (Z/kZ) x Z .
In the following lemma we use the fact that 7Lp (m) = (Z/mZ) x 7Lp to
denote an element of 7%p ( rn ) as x = (r, y) . Also, if n E N, n = r + mk
(o Ç r C rri) then the map n —+ (r, k) imbeds N in 7Lp (m) .
Lemma 1 .
The mapping c,o : 7Lp(m) --} Vq : (r, y) —> ag r (gm)y is a homeomor-
phisrn .
The proof of this lemma can be found in [2, p . 377] .
Corollary .
If g 1 (mod p ) , i . e . m = 1, then the mapping : 7Lp ~Vg : x
	
aq'
is a homeomorphism .
Let ~ be an element of 7L p \{0} . We want to know the valuation of the
p-adic integer (gm)O — 1 . Therefore we need two lemmas :
The following lemmas (2 and 3) are proved in [3] :
Lemma 2.
Let a be an element of 7Lp , a 1 (moda 1 (modr 1 .
If (p, r)
	
(2,1), ,C3 E 71p\{o} then a@
	
1 (modpr+ordp,Q ), 1
(mod pr+ l+vrdp ,Q ) .
Corollary .
Let qm 1 (modpko), qm 1 (modpko+l) . lf (p, k0 ) (2,1 ) , ,C3 E
71p\ {o} then (gm))3 1 (modp"+ordP Q ), (gm)P 1 (modpk°+1+°rdp Q) .
In Lemma 2 we excluded the case where (p, r) = (2, 1) . This case wil l
be handled in the following lemma:
Lemma 3.
Let a be an element of Z2, a 3 (mod 4) . Define a natural number n
by a = 1+2+22 E, E = eo + ~12 +
E222 + . . ., Eo =El = . . . — En_l = 1 ,
En = O .
If ,3 E Z 2\{O}, ord2 ~ = 0 then a0 T 1 (mod 2), aP # 1 (mod 4) .
NORMAL BASES
	
37 5
If i5' E Z 2 \{O}, ord2 fi = k
~
1 then cx~ - 1 (mod2 2+02 ~), a0 # 1
(mod 2n+3+ord2 i5') .
Corollary.
If q 3 (mod 4), we define a natural number N by q = 1 + 2 + 22 e ,
e = ea + e 1 2 + e2 2 2 + . . . = e i = . . . = EN—1 = 1, E N = 0 .
If fl E Z2 \{O}, orde fi = 0 then qQ - 1 (mod 2), q a # 1 (mod 4) .
If Q E Z 2 \{O}, orde fi = k > 1 then q Q 1 (mod2 2+01 ' 2 ~), qp 1
(mod 2N+3+ord e
We remark that is possible to write each x and element of Vq in the
following way : x = agi x (gm )°e x where i x is a natural number, 0 < ix C
m, and where as is an element of Zp . This immediately follows from
Lemma 1 . This leads us to the following definition :
Definition .
We now define a sequence of functions ek in the following way. Write
k(E N) in the form k = i + m j, 0 Ç i C m (i, j E N) . The functions e k
are defined by
ek(x) = e i -I-mj( x ) = 1 if x = agzx (gm )'x where ix = 2, q (xx .
= 0 otherwise.
Let us use the notation B (b, r- ) for the 'open' disc with radius r and with
center b, i .e . B(b, r- ) = {x E Vq l Ix -bi C r}, and B(b,r) far the 'closed '
disc with radius r and with center b, Le . B (b, r) = {x E Vg ! i x— b l Ç r} .
In the following lemmas we will show that the functions e k (x) are
characteristic functions of discs . There exists a ko such that qm EE. 1
(modp c 0) , qm # 1 (modp co) . We distinguish two cases: (p, ko )
(2,1) ( :Lemma 4), and (p, ko ) = (2,1) Le . q - 3 (mod4) (Lemma 5) . If
we use the same notation in Lemmas 4 and 5 as in the definition, we
have
Lemma 4.
Let qm . - 1 (modp'0), qm

1 (modp'0+l) and suppose (p, kp )
(2,1) .
If 0 < i < m then e i (x) is the characteristic function of the closed
disc B (aqi , p-k® ) , and if 0 Ç i C m, j > 1 then e k (x) = e i+jm(x) is the
-,~ -
characteristic function of the open disc B (aq(qm)2,
(P0)
.
376

A. VERDOOD T
Proof:
s
Let j = E be the Henseldevelopment of j E No, with a s different
i=o
from zero .
If we use the notation x = ag ix (gm }°` x (O Ç is C m) far an element x
of Vq , we will show the following :
a) if o Ç i C m : 1 x— agi 1 Ç p-" if and only if i s = i .
b) if o Ç i c m, j
	
1 : Ix - agi (gm )j 1 C Pj ° if and only if =
j q a x .
We first prove a) . If ix = i, then Ix — agi = 1 — ag i =
1x — 11 Ç p-k° by the corollary to Lemma 2 .
If then
Ix -
agz
I=
agZx ( gm}ax _ aqz
= max{ I agzx (gm rx — aq ix 1, Iaqzx — agz 1 } = 1 ,
since laqix (qm)x — agzx C p- k0 , Iaqix — agi = 1 . This proves a) .
Now we prove b) .
Suppose is — i, j
	
as . Then 1x - ag i (gm ) j 1= I(gm)"x-j - - 11 <
p- k °-(s+
1
) by the corollary following Lemma 2, since j is an initial
part of a~ . Since j is strictly smaller than p(8+1) , we conclude tha t
I ¡ 11
°
ix — agzlgmliI
C k
.
k
For the converse, suppose Ix — agi (qm)i < P
j
° . Then we must have
that equals í, since otherwise 1x -- agi (g
m
)j I = 1 :
Ix — agz(gm)a 1 =
lagzx (
qm ) "x -- aqi
(gm) a } 1
= max{ agzx ( gm )"
x
— aqzx 1, I aqix _ aqz 1,
laqi — agz l
gm l
i 1 }
= 1
since laqix (qm)ESx — aqzx < p-'0, 1aqi — agz (gm) 3 I Ç p-k ° (corollary to
Lemma 2) and lag ix — aq i = 1 if is different from i .
So we have 1 ax -j
— 11 < and from this it follows that
1
— 11 Ç p-k°
-
(
s+ 1) since j is at least ps . This means that
ordP (as — j) is at least s + 1 (again by the corollary to Lemma 2) and
so we conclude that j is an initial part of a~ . n
Lemma 5.
lf q 3 (mod 4), with q = 1 + 2 + 2 2e, where s = so + e 1 2 +
s222
+ . . . ,
Eo = E1 = . . . = EN-1 = 1, EN = O,
then eo(x) is the characteristic
NORMAL BASES

377
function of V9 , and ei (x) is the characteristic function of the open disc
B (aqj , (2-'2))
J
Proof:
In this case m equals one and we use the notation x = agax far an
element x of Vq .
It is clear that eo(x) is the characteristic function of Vq .
If j is at least one, we prove : ix — agj ~ < 2
	
	 (N. +2 ) if and only if j q ax .3
Suppose j q cx x . Then lx — aqj I _ le x - i — 11 < 2-(N+2)-(s+1 )
(corollary following Lemma 3), and since j is strictly smaller than 2s+1 ,
we conclude — aqj < 2
-~N+ 2 )
For the converse, suppose lx— aqj < 2	 7 2 ) . Then q~` x Tj — C
2
-(N+2)
and so -1 i C 2-(N+2 ) -(s+1) since j is at least 2S . By the3
corollary to Lemma 3, we have that ord 2 (cxx — j) is at least s + 1 and so
j is an initial part of cxx . ■
Corollary .
The functions (e k (x)) are continuous functions on Vq .
In the following theorem we prove that the sequence (ek (x)} forms a
normal basis for C(Vq -~ K) . This implies that if f is an element of
co
C(Vq -~ K), there exists elements ak of K such that f(x) = >l akek(x)
k= o
where the right-hand-side is uniformly convergent . We are able to give an
expression for the coefficients ak . The proof of this theorem is analogou s
to the proof of Theorem 62.2 in [1] .
Theorem 1 .
The functions (e k (x)) form a normal basis for C(Vq ~ K) . lf f is an
element of C(Vq ---~ K) then f can be written as a uniformly convergent
cc
series f(x) = Eakek(x) where
k= o
ak = f(aqk )
	
if 0 < k < m
ak = ai+im = f(ag2 (4 m)j ) — f(a42 ((Im)J-) if 0 < i c m, j > 0 .
Proof:
Let f be an element of C(I/q ~ K), and let ak be defined as ak =
f(aq k ) if 6 Ç k < m, ak = ai+jm — f(agz(qm)j ) — f(agz(gm )j-) if
0 ÇiCm, j > O .
378

A . VERDOOD T
We first prove that ak tends to zero if k tends to infinity : for all e > 0 ,
there exists a J such that k ~ J implies ia k l < E . To prove this, w e
distinguish two cases :
i) Let qm 1 (modpk°), qm
	
1 (modpk ° +1 ), with (p, ko) (2,1) .
Since the function f is continuous on Vq , it is uniformly continuous
on Vq, and so there exist an S, such that 1x - y i < p- (ko+s ) implies
f (x) - f (y) ~ C E . We then put J = psm .
If k ~ J, and k equals i+ j m with 0 Ç i C m, then we have that j ~ ps
and so (corollary to Lemma 2) I - agi ( gm)i- ! =
(qm)J_a__1 Ç
p -(kQ+s) and this implies that kkI = f (agi (gm ) 3 ) - f(aq(qm ) 3 ) I C E .
ii} Let g 3 (mod 4 ) , q-- 1+2+2 2 e, e= eo+e 1 2 + E222 + . . . ,
EO = El = • • • = eN -1 = 1, EN = O . We remark that m equals one in this
case .
Since the function f is continous on Vq , it is uniformly continuous
on Vq, and so there exist an S, such that ix - y i < 2- ( N+ 2 + s ) implies
~ - f(y ) ~ C E . We then put J = 2s .
If k ~ J, then (corollary to Lemma 3) I qk - qk- I = I qk-k- - 1I Ç
2( 2 +8) and this implies that lak = f (q k ) - f (q k - } 1 C E .
We conclude that ak tends to zero if k tends to infinity.
If f is an element of C (1/g -4 K), we introduce a function g(x) defined
~
by g(x) = > akek(x) with a k as in (*) . Since lia ke k ~I Ç ia k i —> 0 ,
k= 0
the series on the right-hand-side converges uniformly, so the function g
is continous as a uniformly limit of continuous functions . We can prove
that g(aqk) = f (aqk) if O . < k < m and that g(ag i (gm ) j )—g(aq i (qm}j - ) _
f (agZ (gm) 3 )- f (agi (gm)j- ) for 0 Ç i C m, j > 0 . Then we have g(aqk ) =
f(aqk ) for all natural numbers k and by continuity, we conclude that
f (x) = g(x) .
00
So we have f (x) = E a k e k (x), with a k as in (*) .
k-o
It is clear that iif < ó á~x{ia k but we also have f (aq k) ~ IfM and
(agi (qm)i) - f(agi (qm ) i_ )i < IIfM so we conclude f ii = óax{la ki} .
Finally we prove the uniqueness of the coefficients .
00 00 00
If f(x) = E ake k (x) = E b k e k (x), then E (ak - bk )ek (x) = 0. So
k=a k=o k=o
omax{ I a k - b id} = 0, from which it follows . that a k = bk for all k . This
proves the theorem. ■
NORMAL BASES

379
3. More bases for C(Vq K)
We can make more normal bases, using the basis (ek(x)) of Theorem 1 :
Theorem 2.
n
Let (e(x)) be as above, and define rn (x) = E cn , j e j (x), cn, j E K,j=o
cn ; n 0 . Then (r(x)) forms a normal basis for C(Vq ---} K) if and only
if
	
= 1 and iCn ;n! = 1 for all n .
The proof of this theorem will not be given here, since it is analogou s
to the proof of Theorem 2 in [3] .
Reraark.
An analogous result can be found on the space C(7Lp —> K), if we
replace the sequence (e(x)) by the van der Put basis (g(x)) from the
introduction .
We can extend Theorem 2 to the following :
Theorem 3.
Let (rn (x)) be a sequence as found in Theorem 2, which forms a norma l
basis for C(Vq —> K), and let (sn(x)) be a sequence such, that sn(x) =
n
Edm ;jnj (3), dn ;j E K, dn ;n D .
J=o
Then the following are equivalent:
i) (sn(x)} forms a normal basis for C(Vq —> K) .
ii} =1, I dn;nl = 1 .
iii) Idn ;j Ç 1, k41 ;nI = 1 .
Proof:
i) ii} follows from Theorem 2, using the expression
n
E cn ;j ej (x), and ii} <=> iii) follows from the fact that (r(x)) forms a
j=o
normal basis . ■
Exa:mples .
1) If a sequence (r(x)), as found in Theorem 2, forms a normal
basis of C(Vq --> K), then so does (s(x)), where sn (x) = ro (x) +
r l (x) + . . . + rn (x) : apply iii) .
2) If we put for O Ç i C m,
ri (x) = 1 if x = agzx (gm)" where ix =
= 0 otherwise,
rn(x)
380 A . VERDOODT
anf for k ~ m we put
rk (x) = ri+7mj (x) (0 ç i C m) = 1 if x = aqi x (qm )a x
where ix = i, j
= 0 otherwise .
then (r(x)) forms a normal basis for C(Vq --} K). We can apply
iii) since ri(x) = e i (x) for 0 Ç i < m, rk (x) = e i (x) — ek (x) for
k = i+mj, 0 Ç i C m, j> 0 . If f E C(Vq —> K), then there exists
a uniformly convergent expansion of the form f (x) = E ckrk(x) ,
k=0
where
Ck = Ci+jm
= f (ag i (gm ) j- ) — f (ag i ( g m )i ) if 0 Ç i < m, j 0, and
cc,
c i = .f ( agz ) - E ci+jm

if 0 c i C m .
j=1
References
1. W . H . SCHIKHOF, "Ultrametric Calculus : An Introduction to p-adic
Analysis, " Cambridge University Press, 1984 .
2. A . VERDOODT, Jackson's Formula with remainder in p -adic analy-
sis, Indagationes Mathematicae, N.S. 4(3) (1993), 375-384 .
3. A . VERDOODT, Normal bases for Non-Archimedean spaces of con-
tinuous functions, Publicacions Matemàtiques UAB 37 (1993) ,
403-427 .
Vrije Universiteit Brusse l
Faculty of Applied Sciences
Pleinlaan 2
B 1050 Brussels
BELGIUM
Rebut el 9 de Febrer de 1994


Normal bases for the space of continuous functions defined on a subset of $\Bbb Z_p$

Abstract

Similar works

Full text

Available Versions

Revistes Catalanes amb Accés Obert

Normal bases for the space of continuous functions defined on a subset of Zp\Bbb Z_pZp​

Abstract

Similar works

Full text

Available Versions

Revistes Catalanes amb Accés Obert

Normal bases for the space of continuous functions defined on a subset of $\Bbb Z_p$