IJMMS 2003:41, 2619–2633
PII. S0161171203301279
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON THE p-ADIC SPECTRAL ANALYSIS
n
)
AND MULTIWAVELET ON L2 (Qp
WONYONG CHONG, MIN-SOO KIM,
TAEKYUN KIM, and JIN-WOO SON
Received 20 January 2003
To Dong-Sik Kim on his 62nd birthday
We try to generalize an explicit construction for an orthonormal system of eigenn ) which was originally given by
functions of the Vladimirov operator on L2 (Qp
Vladimirov (1988). Also the multiwavelet analysis can be considered as the p-adic
n ), using in part the recent work of Kozyrev (2002).
spectral analysis in L2 (Qp
2000 Mathematics Subject Classification: 11S80, 11M99.
1. Introduction. Let Z, C, Qp , and Cp denote the ring of integers, the field
of complex numbers, the field of p-adic rational numbers, and the completion
of the algebraic closure of Qp , respectively. Let ordp denote the unique p-adic
ordinal over Qp such that ordp (p) = 1. The corresponding non-Archimedian
absolute value is |x|p = p − ordp (x) .
Any p-adic number x ≠ 0 is uniquely represented in the canonical form
∞
x = k=γ(x) xk p k , where γ = γ(x) ∈ Z and xj are integers such that 0 ≤ xk ≤
p − 1, x0 > 0, k = 0, 1, 2, . . . . The fractional part {x}p of a number x ∈ Qp is
defined by



0,


−1
{x}p = 

xk p k ,


if γ ≥ 0 or x = 0,
if γ < 0.
(1.1)
k=γ
It is easy to see from this definition that p γ ≤ {x}p ≤ 1−p γ if γ < 0. The function χp (ξx) = exp(2π i{ξx}p ) for every fixed ξ ∈ Qp is an additive character
of the field Qp and the group Bγ (see [10]). From the relation for fractional
parts we have {x + y}p = {x}p + {y}p − N, N = 0, 1. For later use, we define
the step function Ω(t) by

1, if 0 ≤ t ≤ 1,
Ω(t) =
0, if t > 1.
(1.2)
2620
WONYONG CHONG ET AL.
The space Qpn consists of points x = (x1 , . . . , xn ), xj ∈ Qp , j = 1, 2, . . . , n. The
norm on Qpn is |x|p = max 1≤j≤n |xj |p , xj ∈ Qp . This is a non-Archimedian
norm since |x + y|p ≤ max(|x|p , |y|p ), x, y ∈ Qpn . The space Qpn is clearly
complete metric, locally compact, and totally disconnected space. We introduce the inner product by x, y = x1 y1 + · · · + xn yn , x, y ∈ Qpn .
We denote by Bγn (a) the ball of radius p γ with center at the point a ∈ Qpn
and by Sγn (a) its boundary (sphere), that is,
Bγn (a) = x ∈ Qpn : |x − a|p ≤ p γ ,
Sγn (a) = x ∈ Qpn : |x − a|p = p γ .
(1.3)
For the notational convenience, let Bγn (0) = Bγn and Sγn (0) = Sγn , γ ∈ Z. If a =
(a1 , . . . , an ) ∈ Qpn , then Bγn (a) = Bγ (a1 ) × · · · × Bγ (an ) in Qpn . Clearly, Bγn (a)
and Sγn (a) are closed-open sets.
Recently, very interesting properties of spectral theory in the p-adic number
field were studied (cf. [5, 6, 8, 10]). The important basic operator in the analysis
on complex-valued functions over non-Archimedian local fields K = Qp is the
fractional differential operator D α (see Section 1) introduced and studied by
Vladimirov in [8], and for a general local field K by Kochubei [5]. This operator
considered on L2 (Qp ) has a pure point spectrum with eigenvalues of the infinite multiplicity. An explicit construction of an eigenbasis was first given by
Vladimirov [8]. Due to the infinite multiplicity, it is possible to construct eigenbases with different properties, and a new simpler construction was proposed
recently by Kozyrev (see [6]) and applied by him to the 2-adic interpretation
of wavelets. His result was a motive for our study (see Section 3). A detailed
analysis of the spectrum and eigenfunctions of the Schrödinger-type operator
D α + V (|x|p ) over Qp , with V (r ) → ∞ as r → ∞, is given in [10].
In Section 2, we try to generalize an explicit construction for an orthonormal system of eigenfunctions of the Vladimirov operator on L2 (Qpn ) which was
originally given by Vladimirov [8], and their properties of dimension 1 are
also given in [6]. In Section 3, we give a generalization of Kozyrev’s results
(see Theorem 3.4). Those of multidimensional case with some conditions are
proved. Also the multiwavelet analysis can be considered as the p-adic spectral
analysis in L2 (Qpn ).
2. The Vladimirov operator D α on L2 (Qpn ). We now recall the definitions
and results from the p-adic spectral theory on the p-adic space Qpn (see [2, 4,
5, 6, 8, 9, 10]). The Haar measure dxi (i = 1, 2, . . . , n) is the essentially invariant
measure on the additive group Qp : d(xi +a) = dxi for any a ∈ Qp and so it is
extended up to an invariant measure dx = dx1 dx2 · · · dxn on Qpn in a standard
way, and all integration theories are carried over to Qpn . Normalization is fixed
by taking the measure of Zn
p , the n-times Cartesian product of p-adic integers,
2621
ON THE p-ADIC SPECTRAL ANALYSIS . . .
as being equal to 1:
µ Zn
p =
dx =
Zn
p
|x|p ≤1
dx = 1.
(2.1)
It is now straightforward to calculate the measure of any n-ball and also of
n-sphere, that is, for γ ∈ Z,
µ Bγn =
µ Sγn =
Sγn
dx =
n
Bγ
n
Bγ
dx −
dx = p nγ ,
n
Bγ−1
(2.2)
dx = p nγ 1 −
1
.
pn
(2.3)
A complex-valued function f (x) defined on Qpn is called locally constant if
for any point x ∈ Qpn there exists an integer l(x) ∈ Z such that
f (x + x ) = f (x),
|x |p ≤ p l(x) .
(2.4)
For the set of locally constant functions on Qpn we denote Ᏹ = Ᏹ(Qpn ). We call a
test function for each function in Ᏹ compact support. The set of test functions
which are linear is denoted by Ᏸ = Ᏸ(Qpn ). Let ϕ ∈ Ᏸ. Then there exists l ∈ Z
such that
ϕ(x + x ) = ϕ(x),
x ∈ Bln , x ∈ Qpn .
(2.5)
Such largest number l we call it the parameter of constancy of a function ϕ,
l = l(ϕ). We denote by Ᏸlγ = Ᏸlγ (Qpn ) the set of test functions with support
in the disc Bγn and with parameter of constancy greater than or equal to l. Let
is defined by the formula
ϕ ∈ Ᏸ. Its Fourier-transform F [ϕ] = ϕ
ϕ(ξ)
= F [ϕ](ξ) =
n
Qp
χp ξ, x ϕ(x)dx,
ξ ∈ Qpn .
(2.6)
is the linear isomorphism from Ᏸ onto Ᏸ, and
The Fourier transform ϕ → ϕ
also the inversion formula is valid:
=
ϕ(x) = F −1 [ϕ](x)
n
Qp
χp − x, ξ ϕ(ξ)dξ,
ϕ ∈ Ᏸ.
ϕ,
(2.7)
Lemma 2.1. Let Ᏸ be the set of linear continuous functionals on Ᏸ. Every
function f ∈ L1loc (Qpn ) defines a generalized function f ∈ Ᏸ by
f , ψ =
n
Qp
f (x)ψ(x)dx,
ϕ ∈ Ᏸ.
(2.8)
The Vladimirov operator D α : ψ → D α ψ is defined by the convolution of generalized function f−α and ψ :
D α ψ (x) = f−α ∗ ψ (x)
= f−α (y), ψ(x − y) ,
α ≠ −n ∈ C, x, y ∈ Qpn ,
(2.9)
2622
WONYONG CHONG ET AL.
where f−α (x) = (1 − p α )/(1 − p −α−n )|x|−α−n
. Then
p
D α ψ (x) = F −1 ◦ |ξ|α
p · F [ψ] (x)
=
pα − 1
1 − p −α−n
ψ(x) − ψ(y)
dy.
|x − y|α+n
p
n
Qp
(2.10)
Proof. By (2.2) and (2.3) we calculate the following integrals:
B0n
|x|α−n
dx =
p
= 1−
n
Qp
\B0n
|x|α−n
dx =
p
p γ(α−n)
Sγn
−∞<γ≤0
1
pn
p γ(α−n)
1
pn
p γα =
−∞<γ≤0
1≤γ<∞
= 1−
dx
Sγn
1 − p −n
1 − p −α
(Re α > 0),
(2.11)
dx
p γα = −
1≤γ<∞
1 − p −n
1 − p −α
(Re α < 0),
and we have an analytical continuation for Re α ≥ 0, α ≠ αk = 2kπ i/ ln p, k ∈
Z. Therefore,
n
Qp
|x|α−n
dx =
p
B0n
|x|α−n
dx +
p
n
Qp
\B0n
|x|α−n
dx = 0,
p
(2.12)
where α ≠ αk , k ∈ Z. Then we see that
,ψ =
|x|α−n
p
n
Qp
|x|α−n
ψ(x)dx =
p
n
Qp
|x|α−n
ψ(x) − ψ(0) dx
p
(2.13)
for α ≠ αk , k ∈ Z. Set
f−α (x) =
1 − pα
|x|−α−n
.
p
1 − p −α−n
(2.14)
We obtain
D α ψ (x) = f−α ∗ ψ (x) =
1 − pα , ψ(x − y)
|y|−α−n
p
1 − p −α−n
ψ(x − y) − ψ(x) dy
|y|−α−n
p
=
1 − pα
1 − p −α−n
=
pα − 1
1 − p −α−n
n
Qp
ψ(x) − ψ(x − y)
dy
|y|α+n
p
=
pα − 1
1 − p −α−n
n
Qp
ψ(x) − ψ(y)
dy.
α+n
|x − y|
p
n
Qp
On the other hand, by the formula (2.3) we obtain
1 − pα
−α−n
= |ξ|α
|x|
F f−α (x) = F
p
p,
1 − p −α−n
(2.15)
(2.16)
2623
ON THE p-ADIC SPECTRAL ANALYSIS . . .
for α ≠ αk , k ∈ Z. Since D α ψ = f−α ∗ ψ, we have
F D α ψ = F f−α ∗ ψ = F [f−α ] · F [ψ]
⇐⇒ F D α ψ = |ξ|α
p · F [ψ]
α
⇐⇒ D ψ = F
−1
see [5, 10]
(2.17)
◦ |ξ|α
p · F [ψ].
This completes the proof.
Lemma 2.2 (see [2, 5, 10]). Let χp be the additive character of the field Qp .
Then
n
Bγ
Sγn
Sγn
χp
χp
χp ξ, x dx = p nγ Ω ξp −γ p ;

nγ

1 − p −n ,

p

ξ, x dx = −p n(γ−1) ,



0,
(2.18)
|ξ|p ≤ p −γ ,
|ξ|p = p −γ+1 ,
|ξ|p ≥ p


0,



ξ, x − 1 dx = −p nγ ,



−p nγ 1 − p −n ,
−γ+2
(2.19)
;
|ξ|p ≤ p −γ ,
|ξ|p = p −γ+1 ,
|ξ|p ≥ p
−γ+2
(2.20)
.
Corollary 2.3 (cf. [5, page 37, Proposition 2.3]). Let ξ ∈ Qpn with |ξ|p ≤ 1
and let Re α < 0. Then
|ξ|−α
p =
1 − p −α
1 − p α−n
n
Qp
|ξ|α−n
χp ξ, x − 1 dx.
p
(2.21)
Proof. Since Qpn = ∪γ∈Z Sγn ,
n
Qp
χp ξ, x − 1 dx
|ξ|α−n
p
=
p γ(α−n)
−∞<γ<∞
Sγn
(2.22)
χp ξ, x − 1 dx.
Let |ξ|p = p −k , k ≥ 0. By Lemma 2.2, we see that
−∞<γ<∞
p γ(α−n)
Sγn
χp ξ, x − 1 dx = −p (k+1)α −
p γα 1 − p −n
k+2≤γ<∞
= p kα
1 − p α−n
.
1 − p −α
(2.23)
We therefore obtain the corollary.
Lemma 2.4. For a ∈ Qpn with |a|p > 1, the Vladimirov function
ψ(x) = χp a, x Ω |x|p
(2.24)
2624
WONYONG CHONG ET AL.
is an eigenfunction of the Vladimirov operator
D α ψ(x) = |a|α
p ψ(x).
(2.25)
Proof. Note that, by Lemma 2.1,
D α ψ (x) = F −1 ◦ |ξ|α
p · F [ψ] (x)
=
p α − 1 χp a, x
1 − p −α−n
n
Qp
Ω |x|p − χp a, y − x Ω |y|p
dy,
|x − y|α+n
p
d(cx) = |c|p dx
for c ∈ Qp∗ .
(2.26)
Let |x|p ≤ 1. Then using the fact that every point of a disk is the center of this
disk, we get
n
Qp
Ω |x|p − χp a, y − x Ω |y|p
dy =
|x − y|α+n
p
=
n
Qp
n
Qp
1 − χp a, y − x Ω |y|p
dy
|x − y|α+n
p
1 − χp a, z Ω |z|p
dz.
|z|α+n
p
(2.27)
Let |x|p > 1. We see that
n
Qp
χp a, y − x Ω |y|p
1
dy =
|x − y|α+n
|x|α+n
p
p
=
|y|p ≤1
χp a, y − x dy
χp−1 a, x
|x|α+n
p
|y|p ≤1
(2.28)
χp a, y dy.
By (2.18) of Lemma 2.2 with |a|p > 1, we have
B0n
χp

0,
a, x dx =
1,
if |a|p > 1,
if |a|p ≤ 1.
(2.29)
Therefore, for a, x ∈ Qpn with |x|p > 1 and |a|p > 1,
n
Qp
Ω |x|p − χp a, y − x Ω |y|p
dy
|x − y|α+n
p
=−
n
Qp
χp a, y − x Ω |y|p
dy = 0.
|x − y|α+n
p
(2.30)
2625
ON THE p-ADIC SPECTRAL ANALYSIS . . .
If |z|p > 1, that is, |z|p = p 1 , |z|p = p 2 , . . . , then Ω(|z|p ) = 0. From (2.19) of
Lemma 2.2 we obtain
n
Qp
1 − χp a, z Ω |z|p
dz =
|z|α+n
p
=
|z|p ≥p
1
dz =
|z|α+n
p
1≤γ<∞
p −γ(α+n) p γn 1 −
1≤γ<∞
Sγn
1
pn
|z|−α−n
dz
p
=
1 − p −n
.
pα − 1
(2.31)
Suppose that |z|p ≤ 1, that is, Ω(|z|p ) = 1. Then by (2.19) of Lemma 2.2 and
µ(Sγn ) = p nγ (1 − p −n ), we have
n
Qp
1 − χp a, z Ω |z|p
dz
|z|α+n
p
=
1 − χp a, z dz
p −γ(α+n)
−∞<γ≤0
=
1 − p −n
1 − pα
Sγn

1


p −nγ 1 − n ,



p
−
p γ(α+n) · −p n(−γ−1) ,



0≤γ<∞

0,
(2.32)
|a|p ≤ p γ ,
|a|p = p γ+1 ,
|a|p ≥ p γ+2 .
We now set |a|p = p k , k ≥ 1. Then
n
Qp
1 − χp a, z Ω |z|p
dz
|z|α+n
p

1


p −nγ 1 − n ,



p
1 − p −n
=
−
p γ(α+n) · −p −n(γ+1) ,

1 − p α 0≤γ<∞




0,
k ≤ γ,
k − 1 = γ,
(2.33)
k−2 ≥ γ


1
1 − p −n  γ(α+n) −nγ
(k−1)(α+n) −nk 
.
1− n −p
−
p
p
p
=
1 − pα
p
k≤γ<∞
Hence we have
1 − χp a, z Ω |z|p
p α − 1 χp a, x
dz
n
1 − p −α−n
|z|α+n
Qp
p
1 − p −n 1 − p −n 1 − p −n p kα
pα − 1
(k−1)α−n
−
+p
+
= ψ(x)
1 − p −α−n p α − 1
1 − pα
1 − pα
D α ψ(x) =
= ψ(x)
p α − 1 p kα 1 − p −α−n
1 − p −α−n
pα − 1
= ψ(x)p kα = |a|α
p ψ(x)
(2.34)
that finishes the proof of the lemma.
2626
WONYONG CHONG ET AL.
We will consider a direct product group
G(n) = Qp /Zp × · · · × Qp /Zp .
(2.35)
n times
Then any nonzero element N ∈ G(n) is representable in the form

N = N1 , . . . , Nn
=
m1
−i
N1i p , . . . ,
i=1
m
n

Nni p
−i 
,
(2.36)
i=1
where Nki ∈ {0, 1, . . . , p − 1}, k = 1, . . . , n.
Theorem 2.5. Suppose that Vladimirov functions are as follows:
(n)
ψγ,j,N (x) =
p −γn/2
c |a|p , n
χp
"
!
aj, p γ x Ω p γ x − N p ,
|a|p > 1,
(2.37)
where γ ∈ Z, N ∈ G(n) , j = 1, . . . , p − 1, and a, x ∈ Qpn such that
c |a|p , n =
p n (p − 1)
.
n
|a|n
p p −1
(2.38)
Denote by (·, ·) the inner product in L2 (Qpn ). Set the inner product
"
!
(n)
(n)
ψγ,j,N , ψγ ,j ,N =
p −(γn+γ n)/2
c |a|p , n
n
Qp
"
!
χp − aj, p γ x Ω p γ x − N p
× χp
aj , p γ x
(2.39)
! "
Ω p γ x − N p dx.
Then
!
"
(n)
(n)
ψγ,j,N , ψγ ,j ,N =
1
δγγ δNN δjj ,
c |a|p , n
(2.40)
where δαβ is the Kronecker symbol.
Proof. Without loss of generality, we now assume that γ ≤ γ . If |p γ x|p ≠
|N|p > 1, then by the isosceles triangle principle of the non-Archimedian norm,
!
"
Ω p γ x − N p = 0.
(2.41)
This means that we can set |p γ x|p = |N|p and also denote
!
!
"
"
" ! " ! Ω p γ x − N p Ω p γ x − N p = Ω p γ x − N p Ω p γ −γ N − N p .
(2.42)
2627
ON THE p-ADIC SPECTRAL ANALYSIS . . .
Therefore we have
"
!
(n)
(n)
ψγ,j,N , ψγ ,j ,N =
p −(γn+γ n)/2
c |a|p , n



0,



 p −γn
=
 c |a|p , n





n
Qp
χp
a pγ j − pγ j , x
!
" × Ω p γ x − N p Ω p γ −γ N − N p dx
(2.43)
n
Qp
γ
if γ < γ ,
ap (j − j), x
!
"
×Ω p γ x − N p Ω |N − N |p dx,
χp
if γ = γ .
Since N, N ∈ G(n) , Ω(|N − N |p ) = δNN . Let |a|p > 1. We note that
p −γn
n
Qp
χp
= p −γn
= χp

0,
=
1,
ap γ (j − j), x
|p γ x−N|p ≤1
χp
a(j − j), N
!
"
Ω p γ x − N p dx
ap γ (j − j), x
p ≤1
|x|
χp
dx
dx
a(j − j), x
(2.44)
j ≠ j,
j = j,
using (2.18) of Lemma 2.2. From above it follows the formula
!
"
(n)
(n)
ψγ,j,N , ψγ ,j ,N =
1
δγγ δNN δjj .
c |a|p , n
(2.45)
This completes the proof.
Theorem 2.6. Let the inner product (·, ·), and so forth, be as in Theorem 2.5.
Then
!
"
(n)
Ω |x|p , ψγ,j,N =
p −γn/2
c |a|p , n
θ(γ, a)δN0 ,
(2.46)
where θ(γ, a) = 0 if |a|p ≥ p γ+1 and θ(γ, a) = 1 if |a|p ≤ p γ .
Proof. First, let |p γ x|p ≠ |N|p . Then |p γ x − N|p = max(|p γ x|p , |N|p ).
Since N ∈ G(n) , |N|p ≥ p. Hence we see that
!
"
Ω p γ x − N p = 0
if p γ x p ≠ |N|p .
(2.47)
2628
WONYONG CHONG ET AL.
Next, let |p γ x|p = |N|p , that is, ordp (p γ x) = ordp (N). Set p γ x = (p γ x1 , . . . ,
p γ xn ) and N = (N1 , . . . , Nn ). Then
min
1≤i≤n
ordp p γ xi
= min
1≤i≤n
ordp Ni
.
(2.48)
If |x|p > 1, Ω(|x|p ) = 0, and we assume that |x|p ≤ 1, then it follows the
canonical form
xi = xi0 + xi1 p + · · · + xi(|γ|−1) p |γ|−1 + xi|γ| p |γ| + · · · ,
Ni = p γ Ni|γ| + · · · + p −1 Ni1 ,
(2.49)
i = 1, . . . , n,
where x = (x1 , . . . , xn ) ∈ Zn
p and 0 ≤ xij ≤ p − 1, j = 0, 1, . . . . Then we get
p γ x − N = p γ x1 , . . . , p γ xn − N1 , . . . , Nn
= p γ x10 + · · · + x1|γ| p |γ| + · · · , . . . , p γ xn0 + · · · + xn|γ| p |γ| + · · ·
−
=
p γ N1|γ| + · · · + p −1 N11 , . . . , p γ Nn|γ| + · · · + p −1 Nn1
1(|γ|−1) p −1 + x1|γ| + x1(|γ|+1) p + · · · , . . . ,
10 p γ + · · · + x
x
n(|γ|−1) p −1 + xn|γ| + xn(|γ|+1) p + · · ·
n0 p γ + · · · + x
x
.
(2.50)
i0 ≠ 0, . . . , or x
i(|γ|−1) ≠ 0 for i = 1, . . . , n, then
If x
!
"
Ω p γ x − N p = 0
since p γ x − N p ≥ p.
(2.51)
i(|γ|−1) = 0 for i = 1, . . . , n; we have
i0 = · · · = x
Let x
!
"
Ω p γ x − N p ≠ 0,
(2.52)
that is, p γ x −N = ((x1|γ| +x1(|γ|+1) p +· · · ), . . . , (xn|γ| +xn(|γ|+1) p +· · · )) ∈ B0n .
Hence
!
"
(n)
Ω |x|p , ψγ,j,N =
=
p −γn/2
c |a|p , n
n
Qp
p −γn/2
c |a|p , n
B0n
Ω |x|p χp
"
!
aj, p γ x Ω p γ x − N p dx
χp ajp γ , x dx = 0
(2.53)
2629
ON THE p-ADIC SPECTRAL ANALYSIS . . .
by (2.18) of Lemma 2.2 with γ < 0. Under the assumption that N ≠ 0, we always
obtain
!
"
(n)
Ω |x|p , ψγ,j,N ≡ 0.
(2.54)
On the other hand, now set N = 0, we see that
!
"
(n)
Ω |x|p , ψγ,j,N =
p −γn/2
n
x∈B0n , x∈Bγ
c |a|p , n



0,
p −γn/2
=



c |a|p , n

p −γn/2


,

c |a|p , n
=



0,
x∈B0n
χp
aj, p γ x dx
χp
aj, p γ x
if Bγn
dx,
B0n
if B0n Bγn (2.55)
|a|p ≤ p γ ,
|a|p ≥ p γ+1 .
Therefore, we have
"
!
(n)
Ω |x|p , ψγ,j,N =
p −γn/2
c |a|p , n
θ(γ, a)δN0
(2.56)
for θ(γ, a) = 0 if |a|p ≥ p γ+1 and θ(γ, a) = 1 if |a|p ≤ p γ . This completes the
proof.
Theorem 2.7. The system of Vladimirov function in Theorem 2.5 with |a|p >
1 is a complete orthonormal system of eigenfunctions of Vladimirov operator
D α in L2 (Qpn ).
(n)
Proof. To prove that {ψγ,j,N } is a complete system, it is enough to check
that the Parseval identity for the function Ω(|x|p ) forms an orthonormal basis
in L2 (Qpn ) (see [3, 4]). Set |a|p = p k for k = 1, 2, . . . . By Theorem 2.6, it is obvious
that
$
#
(n)
Ω |x|p , ψγ,j,N
γ,j,N
2
=
∞ p−1
γ=k j=1
p −γn
c |a|p , n
p −kn
p −1
=
= 1,
c |a|p , n 1 − p −n
where
(2.57)
n
c(|a|p , n) = p n (p − 1)/|a|n
p (p − 1). On the other hand, by (2.39),
Ω |x|p , Ω |x|p
which implies our result.
= 1,
(2.58)
2630
WONYONG CHONG ET AL.
Corollary 2.8 (see [6]). The system of Vladimirov function in Theorem 2.7
with a = p −1 and n = 1 is a complete orthonormal system of eigenfunctions of
Vladimirov operator D α in L2 (Qp ).
3. Interpretation of multiwavelets. Multiwavelets have been used in the
data compression, noise reduction, and solution of integral equations. Because multiwavelets are able to offer a combination of orthogonality, symmetry, higher order of approximation, and short support, methods using multiwavelets frequently outperform those using the comparable scale wavelets.
Multiresolution produces an orthonormal basis of wavelets at all scales γ ∈ Z
(cf. [1]).
The wavelet basis in L2 (Rn
+ ) is a basis given by shifts and dilations of the so(n)
called mother wavelet function. For x ∈ Rn
(x)
+ , we define the Haar wavelet Ψ
by
Ψ (n) (x) ≡



1,






−1,






0,
1 n
,
x ∈ 0,
2
n
1
x∈
,1 ,
2
otherwise,
(3.1)
where
[a, b]n = [a, b] × · · · × [a, b]
for a, b ∈ R.
(3.2)
n times
A wavelet basis is a function Ψ (n) ∈ L2 (Rn
+ ) such that
%
Ψγ,N (x) ≡ 2−γn/2 Ψ (n) 2−γ x − N : γ ∈ Z, N ∈ Zn
(n)
&
(3.3)
is a basis for L2 (Rn
+ ) (see [7]).
Remark 3.1. Spline bases have a maximal approximation order with respect to their length; however, spline uniwavelets are only semiorthogonal. A
family of spline multiwavelets that are symmetric and orthogonal is developed
(cf. [1]).
We may define a mapping ρ : Qpn → Rn
+ by

ρ
∞
i1 =γ1
i1
ai1 p , . . . ,
∞
in =γn

ain p
in 


∞
∞
1 −i1
−in 
ai p , . . . ,
ain p
=
,
p i =γ 1
i =γ
1
1
n
n
(3.4)
ON THE p-ADIC SPECTRAL ANALYSIS . . .
2631
where γj ∈ Z and aij ∈ {0, 1, . . . , p −1} for j = 1, . . . , n. This map ρ is clearly not
one to one. The following map is a one-to-one map:
ρ : G(n) → Zn
+,
(3.5)
n
where Zn
+ = N ∪ {0}, N = {1, 2, . . .}.
Lemma 3.2. (1) The map ρ satisfies the estimate
ρ(x) − ρ(y) ≤ √n|x − y|p .
(3.6)
(2) For N ∈ G(n) and m, k ∈ Z, the map ρ satisfies the conditions
n
−m
ρ : p m N + p k Zn
ρ(N) + 0, p −k ,
p → p
−m
n ρ(N) + 0, p −k
ρ : Qpn \ p m N + p k Zn
→ Rn
,
p
+\ p
(3.7)
up to a finite number of points.
(3) The map ρ maps the Haar measure µ on Qpn onto the Lebesgue measure
n
ν on Rn
+ : for measurable subspace X ⊂ Qp ,
µ(X) = ν ρ(X) .
(3.8)
(4) Suppose that
2
n
ρ ∗ : L 2 Rn
+ → L Qp ; f (x) → f ρ(x)
(3.9)
is a unitary operator. Then the map ρ maps the Haar wavelet (3.1) onto the
Vladimirov function ψ(x) in Lemma 2.4 with p = 2 and (2−1 , . . . , 2−1 ) ∈ Q2n ,
that is,
ρ ∗ : Ψ (n) (x) → ψ(x).
(3.10)
Proof. The proof is essentially due to [6, Lemma 3-6].
Corollary 3.3. Let Sγ = {x ∈ Qpn | |x|p = p γ }. Then
ρ p k Zn
p =
'
ρ p k Sγ ,
γ≤0
ρ p k Sγ ∩ ρ p k Sγ 

 1 ,
= p γ+k

∅,
γ = γ + 1,
(3.11)
otherwise.
Theorem 3.4. For p = 2, the map ρ maps the orthonormal basis of wavelets
(n)
in L2 (Rn
+ ) (see (3.3)) onto the basis ψγ,1,N of the eigenfunctions for the Vladimirov
2632
WONYONG CHONG ET AL.
operator in Theorem 2.5 with a = (2−1 , . . . , 2−1 ) ∈ Q2n and |a|2 > 1:
ρ ∗ : Ψγ,ρ(N) (x) → (−1)|N| 2n − 1 ψγ,1,N (x),
(n)
(n)
(3.12)
where |N| = N1 + · · · + Nn .
Proof. From (3.3) and Part (4) of Lemma 3.2, we obtain
Ψγ,ρ(N) ρ(x) = 2−γn/2 Ψ (n) 2−γ ρ(x) − ρ(N)
(n)
= 2−γn/2 Ψ (n) ρ 2γ x − N
= 2−γn/2 ρ ∗ Ψ (n) 2γ x − N
(3.13)
= 2−γn/2 ψ 2γ x − N .
Next, by the definition
(
c(|a|2 , n) with a = (2−1 , . . . , 2−1 ) ∈ Q2n , we have
c |a|2 , n =
1
.
2n − 1
(3.14)
Hence
(n)
ψγ,1,N (x) =
2−γn/2
c |a|2 , n
= (−1)
χ2
|N| −γn/2
2
a, 2γ x Ω 2γ x − N 2
(3.15)
2
n−1
γ
ψ 2 x −N
since ψ(2γ x −N) = χ2 (−a, N)χ2 (a, 2γ x)Ω(|2γ x −N|2 ). The proof now follows directly.
Acknowledgments. The authors express their gratitude to the referees
for their valuable suggestions and comments. This work is supported by the
Kyungnam University Research Fund, 2003. The third author was supported
by Korea Research Foundation Grant KRF-2002-050-C00001.
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[1]
[2]
[3]
[4]
[5]
[6]
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Ser. Mat. 66 (2002), no. 2, 149–158 (Russian).
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[7]
[8]
[9]
[10]
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R. L. Long and D. R. Chen, Biorthogonal wavelet bases on Rd , Appl. Comput.
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Mat. Nauk 43 (1988), no. 5(263), 17–53 (Russian).
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Wonyong Chong: Department of Information and Communication Engineering,
Kyungnam University, Masan 631-701, South Korea
E-mail address: wychong@kyungnam.ac.kr
Min-Soo Kim: Department of Mathematics, Kyungnam University, Masan 631-701,
South Korea
E-mail address: mskim@mail.kyungnam.ac.kr
Taekyun Kim: Institute of Science Education, Kongju National University, Kongju 314701, South Korea
E-mail address: tkim@kongju.ac.kr
Jin-Woo Son: Department of Mathematics, Kyungnam University, Masan 631-701,
South Korea
E-mail address: sonjin@hanma.kyungnam.ac.kr