$p$-Adic Haar multiresolution analysis
p-ADIC HAAR MULTIRESOLUTION ANALYSIS
V. M. SHELKOVICH AND M. SKOPINA
Abstract. In this paper, the notion of p-adic multiresolution analysis
(MRA) is introduced. We use a “natural” refinement equation whose so-
lution (a refinable function) is the characteristic function of the unit disc.
This equation reflects the fact that the characteristic function of the unit
disc is the sum of p characteristic functions of disjoint discs of radius p−1.
The case p = 2 is studied in detail. Our MRA is a 2-adic analog of the real
Haar MRA. But in contrast to the real setting, the refinable function gen-
erating our Haar MRA is periodic with period 1, which never holds for real
refinable functions. This fact implies that there exist infinity many different
2-adic orthonormal wavelet bases in L2(Q2) generated by the same Haar
MRA. All of these bases are constructed. Since p-adic pseudo-differential
operators are closely related to wavelet-type bases, our bases can be inten-
sively used for applications.
1. Introduction
1.1. p-Adic wavelets and pseudo-differential operators. According to
the well-known Ostrovsky theorem, any nontrivial valuation on the field Q is
equivalent either to the real valuation | · | or to one of the p-adic valuations
| · |p. We recall that the field Qp of p-adic numbers is defined as the completion
of the field of rational numbers Q with respect to the non-Archimedean p-adic
norm | · |p. This norm is defined as follows: if an arbitrary rational number
x 6= 0 is represented as x = pγ m
, where γ = γ(x) ∈ Z, and m and n are not
divisible by p, then
(1.1) |x|p = p−γ , x 6= 0, |0|p = 0.
This norm inQp satisfies the strong triangle inequality |x+y|p ≤ max(|x|p, |y|p).
Thus there are two equal in rights universes: the real universes and the
p-adic one. The latter has a specific and unusual properties. Nevertheless,
there are a lot of papers where different applications of p-adic analysis to
physical problems, stochastics, cognitive sciences and psychology are stud-
ied [6]– [10], [13]– [19], [34]– [36] (see also the references therein). In view
of the Ostrovsky theorem such investigations not only have great interest in
Date:
2000 Mathematics Subject Classification. Primary 11F85, 42C40; Secondary 46F10.
Key words and phrases. p-adic multiresolution analysis, p-adic compactly supported
wavelets.
The first author (V. S.) was also supported in part by DFG Project 436 RUS 113/809
and Grant 05-01-04002-NNIOa of Russian Foundation for Basic Research.
http://arxiv.org/abs/0704.0736v1
2 V. M. SHELKOVICH AND M. SKOPINA
itself, but lead to applications and better understanding of similar problems
in usual mathematical physics.
We recall that there exists a p-adic analysis connected with the mapping Qp
into Qp and an analysis connected with the mapping Qp into the field of com-
plex numbers C, there exist two types of p-adic physics models. For the p-adic
analysis related to the mapping Qp → C the operation of partial differentia-
tion is not defined , and as a result, large number of models connected with
p-adic differential equations use pseudo-differential operators and the theory
of p-adic distributions (generalized functions) (see the above mentioned pa-
pers and books). In particular, fractional operators Dα are extensively used
in applications (see fore-quoted papers and especially [34]).
It is well known that the theory of p-adic pseudo-differential operators (in
particular, fractional operators) and equations closely related to wavelet type
bases. It is typical that p-adic compactly supported wavelets are eigenfunc-
tions of p-adic pseudo-differential operators [3]– [5], [16], [17], [18], [20] – [22].
Thus the wavelet theory plays a key role in application of p-adic analysis and
gives a new powerful technique for solving p-adic problems. This theory starts
development only in resent years and has many open problems.
In [20], S. V. Kozyrev constructed the orthonormal compactly supported
p-adic wavelet basis (1.2) in L2(Qp):
(1.2) θγja(x) = p
−γ/2χp
p−1j(pγx− a)
|pγx− a|p
, x ∈ Qp,
j ∈ Jp = {1, 2, . . . , p−1}, γ ∈ Z, a ∈ Ip = Qp/Zp. Kozyrev’s wavelets (1.2) are
eigenfunctions of the Vladimirov fractional operator [34, IX]. Further develop-
ment and generalization of the theory of such type wavelets can be found in
the papers by S. V. Kozyrev [21], [22], A. Yu. Khrennikov, and S. V. Kozyrev
[16], [17], J. J. Benedetto, and R. L. Benedetto [8], and R. L. Benedetto [9].
In [3], the multidimensional p-adic wavelets generated by direct product of
the Kozyrev one-dimensional wavelets were introduced. In [18], a new type of
p-adic multidimensional wavelet basis was introduced:
θ(m)γsa (x) = p
−γ/2χp
s(pγx− a)
|pγx− a|p
, x ∈ Qp,
where s ∈ Jp;m, γ ∈ Z, a ∈ Ip. Here Jp;m = {s = p−m
s0 + s1p + · · · +
sm−1p
: sj = 0, 1, . . . , p − 1; j = 0, 1, . . . , m − 1; s0 6= 0}, m ≥ 1 is
a fixed positive integer. The multidimensional wavelets from [3] are a par-
ticular case of the last wavelets. Moreover, in [3], [18], there were derived
the necessary and sufficient conditions for a class of multidimensional p-adic
pseudo-differential operators (including fractional operator) to have such mul-
tidimensional wavelets as eigenfunctions.
It remains to point out that for pseudo-differential operators from [3], [18]
a “natural” definition domain is the Lizorkin spaces of distributions Φ′(Qnp ),
introduced in [3]. The space Φ′(Qnp ) is invariant under the mentioned above
pseudo-differential operators. Moreover, the above mentioned p-adic wavelets
belong to the Lizorkin space Φ(Qnp ) of test functions. Recall that the usual
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 3
Lizorkin spaces were studied in the excellent papers of P. I. Lizorkin [24], [25]
(see also [29], [30]).
It’s interesting to compare appearing first wavelets in p-adic analysis with
the history of the wavelet theory in real analysis. In 1910 Haar [12] constructed
an orthogonal basis for L2(R) consisting of the dyadic shifts and scales of one
piecewise constant function. A lot of mathematicians actively studied Haar
basis, different kinds of generalizations were introduced, but during almost the
whole century nobody could find another wavelet function (a function whose
shifts and scales form an orthogonal basis). Only in early nineties a method
for construction of wavelet functions appeared. This method is based on the
notion of multiresolution analysis (MRA in the sequel) introduced by Y. Meyer
and S. Mallat [28], [26], [27]. Smooth compactly supported wavelet functions
were found in this way, which has been very important for some engineering
applications. In this paper we introduce MRA in L2(Qp) and present a concrete
MRA for p = 2 being an analog of Haar MRA in L2(R). The same scheme as
in the real setting leads to a Haar basis. It turned out that this Haar basis
coincides with Kozyrev’s wavelet system. However, 2-adic Haar MRA is not
an identical copy of its real analog. In contrast to Haar MRA in L2(R), we
proved that there exist infinity many different Haar orthogonal bases in L2(Q2)
generated by the same MRA.
1.2. Contents of the paper. In Sec. 2, we recall some facts from the p-adic
theory of distributions [11], [32], [33], [34]. In Sec. 3, some facts from the
theory of the p-adic Lizorkin spaces [3] are recalled.
In Sec. 4, by Definition 4.1 we introduce the MRA adapted to the p-adic
case. In Subsec. 4.2, we introduce the refinement equation (4.7)
φ(x) =
, x ∈ Qp,
whose solution φ(x) = Ω
is the characteristic function of the unit disc,
where where Ω(t) is the characteristic function of the interval [0, 1]. The con-
jecture to use the above equation as the refinement equation was proposed
in [18]. The above refinement equation is natural and reflects the fact that the
characteristic function Ω
of the unit disc B0 is represented as a sum of p
pieces characteristic functions of the disjoint discs B−1(r), r = 0, 1, . . . , p − 1
(see (2.7)).
In Subsec. 4.3, the 2-adic MRA is constructed. Namely, we proved that
MRA is generated by a refinable function which is the characteristic function
φ(x) = Ω
of the unit disc B0 = {x : |x|2 ≤ 1} ⊂ Q2 and satisfies the
refinement equation (4.8)
φ(x) =
, x ∈ Q2.
4 V. M. SHELKOVICH AND M. SKOPINA
By our MRA we construct 2-adic orthonormal wavelet basis (4.15) in L2(Q2),
which is the Kozyrev basis (1.2) for the case p = 2. It turned out that the
Kozyrev wavelet basis is not unique orthonormal wavelet basis.
In Sec. 5, infinity many different 2-adic wavelet orthonormal bases in L2(Q2)
are constructed. Namely, using Theorem 5.1, we construct wavelet functions
ψ(s)(x), s ∈ N whose dilatations and shifts form 2-adic orthonormal wavelet
bases in L2(Q2).
Since many p-adic models use pseudo-differential operators, in particular,
fractional operator, these results on p-adic wavelets can be intensively used in
applications. Moreover, p-adic wavelets can be used to construct solutions of
linear and semi-linear pseudo-differential equations [5], [23].
2. p-Adic distributions
We recall some facts from the theory of p-adic distributions (generalized
functions). Here and in what follows, we shall systematically use the notations
and results from [34] and [11, Ch.II]. Let N, Z, C be the sets of positive
integers, integers, complex numbers, respectively, and N0 = {0} ∪ N. Denote
by Q∗p = Qp \ {0} the multiplicative group of the field Qp.
The canonical form of a p-adic number x 6= 0 is
(2.1) x = pγ(x0 + x1p + x2p
2 + · · · ),
where γ = γ(x) ∈ Z, xj = 0, 1, . . . , p − 1, x0 6= 0, j = 0, 1, . . . . The series
is convergent in the p-adic norm (1.1), and one has |x|p = p−γ. By means of
representation (2.1), the fractional part {x}p of a number x ∈ Qp is defined as
follows
(2.2) {x}p =
0, if γ(x) ≥ 0 or x = 0,
pγ(x0 + x1p + x2p
2 + · · ·+ x|γ|−1p|γ|−1), if γ(x) < 0.
The function
(2.3) χp(ξx) = e
2πi{ξx}p
for every fixed ξ ∈ Qp is an additive character of the field Qp.
According to [34, III.2.], any multiplicative character π of the field Qp can
be represented as
= πα(x) = |x|α−1p π1(x), x ∈ Q∗p,
where π(p) = p1−α and π1(x) is a normed multiplicative character such that
π1(x) = π1(|x|px), π1(p) = π1(1) = 1, |π1(x)| = 1. We denote π0 = |x|−1p .
The space Qnp = Qp × · · · × Qp consists of points x = (x1, . . . , xn), where
xj ∈ Qp, j = 1, 2 . . . , n, n ≥ 2. The p-adic norm on Qnp is
(2.4) |x|p = max
1≤j≤n
|xj|p, x ∈ Qnp ,
where |xj |p id defined by (1.1).
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 5
Denote by Bnγ (a) = {x ∈ Qnp : |x − a|p ≤ pγ} the ball of radius pγ with the
center at a point a = (a1, . . . , an) ∈ Qnp and by Snγ (a) = {x ∈ Qnp : |x − a|p =
pγ} = Bnγ (a) \ Bnγ−1(a) its boundary (sphere), γ ∈ Z. For a = 0 we set
Bnγ (0) = B
γ and S
γ (0) = S
γ . For the case n = 1 we will omit the upper index
n. It is clear that
(2.5) Bnγ (a) = Bγ(a1)× · · · × Bγ(an),
where Bγ(aj) = {xj : |xj − aj |p ≤ pγ} ⊂ Qp is a disc of radius pγ with the
center at a point aj ∈ Qp, j = 1, 2 . . . , n.
Any two balls in Qnp either are disjoint or one contains the other. Every
point of the ball is its center.
According to [34, I.3,Examples 1,2.], the disc Bγ is represented by the sum
of pγ−γ
disjoint discs Bγ′(a), γ
′ < γ:
(2.6) Bγ = Bγ′ ∪ ∪aBγ′(a),
where a = 0 and a = a−rp
−r + a−r+1p
−r+1 + · · ·+ a−γ′−1p−γ
′−1 are the centers
of the discs Bγ′(a), r = γ, γ − 1, γ − 2, . . . , γ′ + 1, 0 ≤ aj ≤ p− 1, a−r 6= 0.
In particular, the disc B0 is represented by the sum of p disjoint discs
(2.7) B0 = B−1 ∪ ∪p−1r=1B−1(r),
where B−1(r) = {x ∈ S0 : x0 = r} = r + pZp, r = 1, . . . , p− 1; B−1 = {|x|p ≤
p−1} = pZp; and S0 = {|x|p = 1} = ∪p−1r=1B−1(r). Here all the discs are disjoint.
We call coverings (2.6) and (2.7) the canonical covering of the discs B0 and
Bγ , respectively.
On Qp there exists the Haar measure, i.e., a positive measure dx invariant
under shifts, d(x + a) = dx, and normalized by the equality
|ξ|p≤1 dx = 1.
The invariant measure dx on the field Qp is extended to an invariant measure
dnx = dx1 · · · dxn on Qnp in the standard way.
If f is an integrable function on Qp, then [11, Ch.II,§2.2], [34, IV]:
(2.8)
dx = pγ ,
f(x) dx =
f(x) dx,
f(x) dx =
f(x) dx−
f(x) dx.
A complex-valued function f defined on Qnp is called locally-constant if for
any x ∈ Qnp there exists an integer l(x) ∈ Z such that
f(x+ y) = f(x), y ∈ Bnl(x).
Let E(Qnp ) and D(Qnp ) be the linear spaces of locally-constant C-valued func-
tions on Qnp and locally-constant C-valued functions with compact supports
6 V. M. SHELKOVICH AND M. SKOPINA
(so-called test functions), respectively [34, VI.1.,2.]. If ϕ ∈ D(Qnp ), according
to Lemma 1 from [34, VI.1.], there exists l ∈ Z, such that
ϕ(x+ y) = ϕ(x), y ∈ Bnl , x ∈ Qnp .
The largest of such numbers l = l(ϕ) is called the parameter of constancy of
the function ϕ. Let us denote by DlN(Qnp ) the finite-dimensional space of test
functions from D(Qnp ) having supports in the ball BnN and with parameters
of constancy ≥ l [34, VI.2.]. The following embedding holds: DlN (Qnp ) ⊂
Dl′N ′(Qnp ), N ≤ N ′, l ≥ l′. Thus D(Qnp ) = lim indN→∞ lim indl→−∞DlN(Qnp ).
The space D(Qnp ) is a complete locally convex vector space.
According to [34, VI,(5.2’)], any function ϕ ∈ DlN(Qnp ) is represented in the
following form
(2.9) ϕ(x) =
pn(N−l)∑
ϕ(cν)∆l(x− cν), x ∈ Qnp ,
where ∆l(x − cν) are the characteristic functions of the disjoint balls Bl(cν),
and the points cν = (cν1 , . . . c
n) ∈ BnN do not depend on ϕ.
Denote by D′(Qnp ) the set of all linear functionals on D(Qnp ) [34, VI.3.].
Let us introduce in D(Qnp ) a canonical δ-sequence δk(x) = pnkΩ(pk|x|p), and
a canonical 1-sequence ∆k(x) = Ω(p
−k|x|p), k ∈ Z, x ∈ Qnp , where
(2.10) Ω(t) =
1, 0 ≤ t ≤ 1,
0, t > 1.
Here ∆k(x) is the characteristic function of the ball B
k . It is clear [34, VI.3.,
VII.1.] that δk → δ, k → ∞ in D′(Qnp ) and ∆k → 1, k → ∞ in E(Qnp ).
The Fourier transform of ϕ ∈ D(Qnp ) is defined by the formula
F [ϕ](ξ) =
χp(ξ · x)ϕ(x) dnx, ξ ∈ Qnp ,
where χp(ξ · x) = χp(ξ1x1) · · ·χp(ξnxn) = e2πi
j=1{ξjxj}p; ξ · x is the scalar
product of vectors.
The Fourier transform is a linear isomorphism D(Qnp ) into D(Qnp ). Moreover,
according to [32, Lemma A.], [33, III,(3.2)], [34, VII.2.],
(2.11) ϕ(x) ∈ DlN(Qnp ) iff F
(ξ) ∈ D−N−l (Q
We define the Fourier transform F [f ] of a distribution f ∈ D′(Qnp ) by the
relation [34, VII.3.]:
(2.12) 〈F [f ], ϕ〉 = 〈f, F [ϕ]〉, ∀ϕ ∈ D(Qnp ).
Let A be a matrix and b ∈ Qnp . Then for a distribution f ∈ D′(Qnp ) the
following relation holds [34, VII,(3.3)]:
(2.13) F [f(Ax+ b)](ξ) = | detA|−1p χp
−A−1b · ξ
F [f(x)]
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 7
where detA 6= 0. According to [34, IV,(3.1)],
(2.14) F [∆k](x) = δk(x), k ∈ Z, x ∈ Qnp .
In particular, F [Ω(|ξ|p)](x) = Ω(|x|p).
The convolution f ∗ g for distributions f, g ∈ D′(Qnp ) is defined (see [34,
VII.1.]) as
(2.15) 〈f ∗ g, ϕ〉 = lim
〈f(x)× g(y),∆k(x)ϕ(x+ y)〉
if the limit exists for all ϕ ∈ D(Qnp ), where f(x) × g(y) is the direct product
of distributions. If for distributions f, g ∈ D′(Qnp ) the convolution f ∗ g exists
then [34, VII,(5.4)]
(2.16) F [f ∗ g] = F [f ]F [g].
Definition 2.1. Let πα be a multiplicative character of the field Qp. A dis-
tribution f ∈ D′(Qnp ) is called homogeneous of degree πα if for all ϕ ∈ D(Qnp )
and t ∈ Q∗p we have the relation
, . . . ,
= πα(t)|t|np
f, ϕ(x1, . . . , xn)
i.e., f(tx) = f(tx1, . . . , txn) = πα(t)f(x), x = (x1, . . . , xn) ∈ Qnp . A homoge-
neous distribution of degree πα(t) = |t|α−1p (α 6= 0) is called homogeneous of
degree α− 1.
3. The p-adic Lizorkin spaces
Let us introduce the p-adic Lizorkin space of test functions
Φ(Qnp ) = {φ : φ = F [ψ], ψ ∈ Ψ(Qnp )},
where
Ψ(Qnp ) = {ψ(ξ) ∈ D(Qnp ) : ψ(0) = 0}.
Here Ψ(Qnp ),Φ(Q
p ) ⊂ D(Qnp ). The space Φ(Qnp ) is called the p-adic Lizorkin
space of test functions . The space Φ(Qnp ) can be equipped with the topology
of the space D(Qnp ) which makes Φ a complete space.
In view of (2.11), the following lemma holds.
Lemma 3.1. ( [3], [4]) (a) φ ∈ Φ(Qnp ) iff φ ∈ D(Qnp ) and
(3.1)
φ(x) dnx = 0.
(b) φ ∈ DlN(Qnp ) ∩ Φ(Qnp ), i.e.,
φ(x) dnx = 0, iff ψ = F−1[φ] ∈
D−N−l (Qnp ) ∩Ψ(Qnp ), i.e., ψ(ξ) = 0, ξ ∈ Bn−N .
Unlike the classical Lizorkin space, any function ψ(ξ) ∈ Φ(Qnp ) is equal to
zero not only at ξ = 0 but in a ball Bn ∋ 0, as well.
Let Φ′(Qnp ) denote the topological dual of the space Φ(Q
p ). We call it the
p-adic Lizorkin space of distributions .
8 V. M. SHELKOVICH AND M. SKOPINA
By Ψ⊥ and Φ⊥ we denote the subspaces of functionals in D′(Qnp ) orthogonal
to Ψ(Qnp ) and Φ(Q
p ), respectively. Thus Ψ
⊥ = {f ∈ D′(Qnp ) : f = Cδ, C ∈ C}
and Φ⊥ = {f ∈ D′(Qnp ) : f = C, C ∈ C}.
Proposition 3.1. ( [3])
Φ′(Qnp ) = D′(Qnp )/Φ⊥, Ψ′(Qnp ) = D′(Qnp )/Ψ⊥.
The space Φ′(Qnp ) can be obtained from D′(Qnp ) by “sifting out” constants.
Thus two distributions in D′(Qnp ) differing by a constant are indistinguishable
as elements of Φ′(Qnp ).
Similarly to (2.12), we define the Fourier transform of distributions f ∈
Φ′×(Q
p ) and g ∈ Ψ′×(Qnp ) by the relations:
(3.2)
〈F [f ], ψ〉 = 〈f, F [ψ]〉, ∀ψ ∈ Ψ(Qnp ),
〈F [g], φ〉 = 〈g, F [φ]〉, ∀φ ∈ Φ(Qnp ).
By definition, F [Φ(Qnp )] = Ψ(Q
p ) and F [Ψ(Q
p )] = Φ(Q
p ), i.e., (3.2) give well
defined objects.
4. Construction of multiresolution analysis
4.1. p-Adic multiresolution analysis. Denote the factor group Qp/Zp by
Ip, i.e.
Ip = {a = p−γ
a0 + a1p+ · · ·+ aγ−1pγ−1
(4.1) γ ∈ N; aj = 0, 1, . . . , p− 1; j = 0, 1, . . . , γ − 1}.
It is well known that Qp = B0 ∪ ∪∞γ=1Sγ, where Sγ = {x ∈ Qp : |x|p = pγ}.
In view of (2.1), x ∈ Sγ , γ ≥ 1 if and only if x = x−γp−γ + x−γ+1p−γ+1 + · · ·+
−1 + ξ, where ξ ∈ B0. Since x−γp−γ + x−γ+1p−γ+1 + · · ·+ x−1p−1 ∈ Ip, we
have a “natural” decomposition of Qp to a union of mutually disjoint discs:
Qp = ∪a∈IpB0(a).
So, Ip is a “natural” group of shifts for Qp.
Definition 4.1. A collection of closed spaces Vj ⊂ L2(Qp), j ∈ Z is called a
multiresolution analysis (MRA) in L2(Qp) if the following axioms hold
(a) Vj ⊂ Vj+1 for all j ∈ Z;
(b) ∪j∈ZVj is dense in L2(Qp);
(c) ∩j∈ZVj = {0};
(d) f(·) ∈ Vj ⇐⇒ f(p−1·) ∈ Vj+1 for all j ∈ Z;
(e) there a function φ ∈ V0 such that the system φ(x − a), a ∈ Ip, form an
orthonormal basis for V0.
The function φ from axiom (e) is called scaling or refinable. It follows im-
mediately from axioms (d) and (e) that the functions pj/2φ(p−j · −a), a ∈ Ip,
form an orthonormal basis for Vj.
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 9
According to the standard scheme (see, e.g., [31, §1.3]) for construction of
MRA-based wavelets, for each j, we define a space Wj (wavelet space) as the
orthogonal complement of Vj in Vj+1, i.e.,
(4.2) Vj+1 = Vj ⊕Wj , j ∈ Z,
where Wj ⊥ Vj , j ∈ Z. It is not difficult to see that
(4.3) f ∈ Wj ⇐⇒ f(p−1·) ∈ Wj+1, for all j ∈ Z
and Wj ⊥Wk, j 6= k. Taking into account axioms (b) and (c), we obtain
(4.4) ⊕j∈ZWj = L2(Qp) (orthogonal direct sum).
If now we find a function ψ ∈ W0 such that the system ψ(x−a), a ∈ Ip, form
an orthonormal basis for W0, then the system p
j/2ψ(p−j · −a), a ∈ Ip, is an
orthonormal basis for L2(Qp). Such a function ψ is called a wavelet function
and the basis is a wavelet basis.
4.2. p-Adic refinement equation. Let φ be a refinable function for a MRA.
As was mentioned above, the system p1/2φ(p−1 · −a), a ∈ Ip, is a basis for V1.
It follows from axoim (a) that
(4.5) φ =
αaφ(p
−1 · −a), αa ∈ C.
We see that the function φ is a solution of a special kind of functional equation.
Such equations are called refinement equations. Investigation of refinement
equations and their solutions is the most difficult part of wavelet theory in real
analysis.
A natural way for construction of a MRA (see, e.g., [31, §1.2]) is the fol-
lowing. We start with an appropriate function φ whose integer shifts form
an orthonormal system, and set V0 = span
: a ∈ Ip
and Vj =
p−jx− a
: a ∈ Ip
, j ∈ Z. It is clear that axioms (d) and (e) of
Definition 4.1 are fulfilled.
Of course, not any such a function φ provides axiom (a). In the real setting,
the relation V0 ⊂ V1 holds if and only if the refinable function satisfies a
refinement equation. Situation is different in p-adics. Generally speaking,
a refinement equation (4.5) does not imply the including property V0 ⊂ V1.
Indeed, we need all the functions φ(· − b), b ∈ Ip, to belong to the space V1,
i.e., the equalities φ(x − b) =
a∈Ip αa,bφ(p
−1x − a) should be fulfilled for all
b ∈ Ip. Since p−1b+ a is not in Ip in general, we can not state that refinement
equation (4.5) implies φ(x − b) =
a∈Ip αa,bφ(p
−1x − p−1b − a) ∈ V1 for all
b ∈ Ip.
The refinement equation reflects some “self-similarity”. The structure of the
space Qp has a natural “self-similarity” property which is given by formulas
(2.6), (2.7). By (2.7), the characteristic function ∆0(x) = Ω
of the unit
10 V. M. SHELKOVICH AND M. SKOPINA
disc B0 is represented as a sum of p characteristic functions of the disjoint
discs B−1(r), r = 0, 1, . . . , p− 1, i.e.,
(4.6) ∆0(x) =
, x ∈ Qp.
Thus, in p-adics, we have a natural refinement equation (4.5):
(4.7) φ(x) =
, x ∈ Qp,
whose solution is φ(x) = ∆0(x) = Ω
. This equation is an analog of the
refinement equation generating Haar MRA in real analysis.
4.3. Construction of 2-adic Haar multiresolution analysis. Now, using
the refinement equation (4.7) for p = 2
(4.8) φ(x) = φ
, x ∈ Q2,
and its solution, the refinable function φ(x) = ∆0(x) = Ω
, we construct
2-adic multiresolution analysis.
(4.9) V0 = span
: a ∈ I2
(4.10) Vj = span
2−jx− a
: a ∈ I2
, j ∈ Z.
It is clear that axioms (d) and (e) of Definition 4.1 are fulfilled and the system
2j/2φ(2−j · −a), a ∈ Ip is an orthonormal basis for Vj, j ∈ Z.
Note that the characteristic function of the unit disc Ω
has a wonderful
feature: Ω(| · +ξ|2) = Ω(| · |2), for all ξ ∈ Z2 because the p-adic norm is non-
Archimedean. In particular, Ω(| · ±1|2) = Ω(| · |2), i.e.,
(4.11) φ(x± 1) = φ(x), ∀ x ∈ Q2.
Thus φ is periodic with the period 1.
In view of this fact, taking into account that 2−1b+ a ( mod 1) is in I2, for
all a, b ∈ I2, it follows from the refinement equation (4.8) that V0 ⊂ V1. By
(4.10), this yields axiom (a).
Due to the refinement equation (4.8), we obtain that Vj ⊂ Vj+1, i.e., the
axiom (a) from Definition 4.1 holds.
Lemma 4.1. The axiom (b) of Definition 4.1 holds, i.e., ∪j∈ZVj = L2(Q2).
Proof. According to (2.9), any function ϕ ∈ D(Q2) belongs to one of the spaces
DlN(Q2), and consequently, is represented in the form
(4.12) ϕ(x) =
pN−l∑
ϕ(cν)∆l(x− cν), x ∈ Q2,
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 11
where ∆l(· − cν) are the characteristic functions of the mutually disjoint discs
ν) ⊂ Q2, cν ∈ BN , ν = 1, 2, . . . pN−l; l = l(ϕ), N = N(ϕ). Since
∆l(x − cν) = Ω(p−l|x − cν |p) = Ω(|plx − plcν |p) and any number plcν can be
represented in the form plcν = aν + bν , where aν ∈ I2, bν ∈ Z2, we have
∆l(x− cν) = ∆l(x− aν). Thus any function ϕ ∈ D(Q2) can be represented in
the form
(4.13) ϕ(x) =
pN−l∑
αν∆l(x− aν), x ∈ Q2, aν ∈ I2, αν ∈ C.
Consequently, on the basis of (4.10), ϕ(x) ∈ V−l. Thus any test function ϕ
belongs to one of the space Vj , where j = j(ϕ).
Since the space D(Q2) is dense in L2(Q2) [34, VI.2], approximating any
function from L2(Q2) by test functions (4.13), we prove our assertion. �
Lemma 4.2. The axiom (c) of Definition 4.1 holds, i.e., ∩j∈ZVj = {0}.
Proof. Suppose that ∩j∈ZVj 6= {0}. Then there exists a function f ∈ Vj for all
j ∈ Z. Hence, due to (4.10), f(x) =
a∈I2 cjaφ
2−jx− a
for all j ∈ Z.
Let x = 2−N(x0+x12+x22
2+· · · ). Since 2−jx = 2−N−j(x0+x12+x222+· · · ),
for all j ≤ −N , we have 2−jx ∈ Z2, and, consequently, |2−jx − a|2 > 1 for
all a ∈ I2, a 6= 0. Thus φ
2−jx − a
= 0 for all j ≤ −N and a ∈ I2, a 6= 0.
Since |2−jx|2 ≤ 1, we have f(x) = cj0 for all j ≤ −N . Similarly, for another
x′ = 2−N
(x′0 + x
12 + x
2 + · · · ), we have f(x′) = cj′0 for all j ≤ −N ′. This
yields that f(x) = f(x′). Consequently, f(x) ≡ C, where C is a constant.
However, if C 6= 0, f 6∈ L2(Q2). Thus, C = 0 and the proof of the theorem is
complete. �
According to the above scheme, we introduce the spaceW0 as the orthogonal
complement of V0 in V1.
(4.14) ψ(0)(x) = φ
Lemma 4.3. The shift system ψ(0)(x− a), a ∈ I2, is an orthonormal basis of
the space W0.
Proof. Let us prove that W0 ⊥ V0. It follows from (4.8), (4.14) that
ψ(0)(x− a), φ(x− b)
ψ(0)(x− a)φ(x− b) dx
for all a, b ∈ I2. Let a 6= b. Since it is impossible a 6= b+ 1, b 6= a + 1, taking
into account that the functions 21/2φ(2−1 · −c), c ∈ I2 are orthonormal, we
obtain
ψ(0)(x − a), φ(x− b)
= 0. If a = b, again due to the orthonormality
12 V. M. SHELKOVICH AND M. SKOPINA
of the system 21/2φ(2−1 · −c), c ∈ I2, taking into account that a2 ,
∈ I2,
we have
ψ(0)(x− a), φ(x− a)
dx = 0.
Thus, ψ(0)(x+ a) ⊥ φ(x+ b) for all a, b ∈ I2.
The refinement equation (4.8) and relation (4.14) imply that
x− 2a
+ ψ(0)
x− 2a
, a ∈ I2.
Since {21/2φ(2−1x − a) : a ∈ I2} is a basis for V1, we have V1 = V0 ⊕W0, i.e.,
(4.2) holds. �
Thus we prove that the collection {Vj : j ∈ Z} is a MRA in L2(Q2) and
the function ψ(0) defined by (4.14) is a wavelet function. This MRA is a 2-
adic analog of the real Haar MRA and the wavelet basis generated by ψ(0) is
an analog of real Haar wavelet basis. But in contrast to the real setting, the
refinable function φ generating our Haar MRA is periodic with the period 1
(see (4.11)), which never holds for real refinable functions. It will be shown
bellow that due of this specific property of φ, there exist infinity many different
orthonormal wavelet bases in the same Haar MRA (see Sec. 5).
Due to (2.3), (2.7), the function ψ(0) can be rewritten in the form ψ(0)(x) =
−1x)Ω(|x|2) and the Haar wavelet basis is
ψ(0)γa (x) = 2
−γ/2ψ(0)(2γx− a)
(4.15) = 2−γ/2χ2
2−1(2γx− a)
|2γx− a|2
, x ∈ Q2, γ ∈ Z, a ∈ I2.
It is clear that
(4.16)
ψ(0)γa (x) dx = 0,
and, according to Lemma 3.1, ψ
γa (x) belongs to the Lyzorkin space Φ(Q2).
Remark 4.1. The Haar wavelet basis (4.15) coincides with Kozyrev’s wavelet
basis (1.2) for the case p = 2. In present paper we restrict ourself by con-
structing the Haar wavelets only for p = 2. Since Haar refinement equation
(4.7) was presented for all p, a similar construction may be easily realized in
the general case. Moreover, it is not difficult to see that Kozytev’s wavelet
function θj(x) from (1.2) can be expressed in terms of the refinable function
φ(x) as
(4.17) θj(x) = χp(p
−1jx)Ω
= p−1/2
, x ∈ Qp,
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 13
where hr = p
2πi{ jr
}p, r = 0, 1, . . . , p− 1, j = 1, 2, . . . , p− 1.
Remark 4.2. In view of periodicity (4.11) of the refinable function φ, one can
use shifts ψ(0)(·+ a), a ∈ I2, instead of shifts ψ(0)(· − a), a ∈ I2.
Now we show that there is another function ψ(1)(x) whose shifts form an
orthonormal basis in W0. Indeed, taking into account (4.11), we have
ψ(1)(x) =
(4.18) =
an its shifts
(4.19) =
(4.20) =
Since the system of functions {φ(2−1x − a) : a ∈ I2} is orthonormal, in
view of (4.11), formulas (4.18)–(4.20) imply that the function ψ(1)(x) and the
function ψ(1)(x − a) are orthonormal, whenever a ∈ I2, a 6= 0, 12 . Here we
take into account that all shifts (up to mod 1) of refinable function in (4.18),
(4.20) are distinct.
Similarly, by (4.18), (4.19), we have
ψ(1)(x), ψ(1)(x+ 2−1)
ψ(1)(x)ψ(1)(x+ 2−1) dx
= 2−1
dx = 0.
ψ(1)(x), ψ(1)(x)
= 2−1
dx = 1.
14 V. M. SHELKOVICH AND M. SKOPINA
Thus all shifts of ψ(1) are orthonormal.
It is clear that the functions (4.18) and (4.19) can be rewritten in the form
(4.21) ψ(1)(x) =
− ψ(0)
+ ψ(0)
It follows that
ψ(0)(x) =
+ ψ(1)
Since the system ψ(0)(· − a), a ∈ I2, forms an orthonormal basis for W0, the
system ψ(1)(· − a), a ∈ I2, is another orthonormal basis for W0.
So, we showed that a wavelet basis generated by the Haar MRA is not
unique.
5. Description of 2-adic Haar bases
5.1. Complex wavelets. Using the fact that all dilatations and shifts (x →
2γx+ a, a ∈ I2) of the Haar wavelet function ψ(0) form a orthonormal basis in
L2(Q2), we show that there exist infinitely many wavelet functions ψ(s), s ∈ N
in W0.
In what follows, we shall write the 2-adic number a = 2−s
a0 + a12 + · · ·+
as−12
s−1) ∈ I2, aj = 0, 1, j = 0, 1, . . . , s − 1 briefly as a rational number
a = m
, where m = a0 + a12 + · · ·+ as−12s−1.
Since the characteristic function of the unit disc φ(x) = ∆0(x) = Ω
periodic with the period ξ ∈ S0, the wavelet function ψ0(x) has the following
evident and important property:
(5.1) ψ(0)(x+ ξ) = −ψ(0)(x), ξ ∈ S0.
Here ξ = 1 + ξ12 + ξ22
2 + · · · , where ξj = 0, 1; j ∈ N.
Before we prove a general result, we consider the simplest particular case.
Consider the function
(5.2) ψ(1)(x) = α0ψ
+ α1ψ
, α0, α1 ∈ C,
and solve the problem when all shifts of this function generates an orthonormal
basis ψ(1)(x+ a), a ∈ I2 in W0.
Taking into account orthonormality of the system ψ(0)(x + a), a ∈ I2 and
relation (5.1), we can see that the function ψ(1)(x) and the functions ψ(1)(x+a)
are orthonormal for all a ∈ I2, a 6= 0, 12 . Thus, in view of (5.1), the system
of functions ψ(1)(x + a), a ∈ I2 is orthonormal if and only if the system of
functions (5.2) and
(5.3) ψ(1)
= −α1ψ(0)
+ α0ψ
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 15
is orthonormal. Hence, we have |α0|2 + |α1|2 = 1. In other words, the matrix
α0 α1
−α1 α0
is unitary. Thus, the function (5.2), where |α0|2 + |α1|2 = 1 is the wavelet
function. It is clear that the wavelet function (4.21) is a particular case of the
wavelet function (5.2).
Consequently, all dilatations and shifts of ψ(1)(x) form 2-adic orthonormal
wavelet basis in L2(Q2).
Now we will prove a general theorem.
Theorem 5.1. Let s = 1, 2, . . . . The function
(5.4) ψ(s)(x) =
2s−1∑
is the wavelet function (whose dilatations and shifts form 2-adic orthonormal
wavelet basis in L2(Q2)) if and only if
(5.5) αk = 2
−s(−1)k
2s−1∑
−iπ 2r+1
k, k = 0, 1, 2, . . . , 2s − 1,
γk ∈ C, |γk| = 1.
Proof. Suppose that ψ(s)(x), s ≥ 1 is given by formula (5.4). Since the system
ψ(0)(· + a), a ∈ I2 is orthonormal (see Subsec. 4.3) and in view of relation
(5.1), it is easy to see that ψ(s) and ψ(s)(·+ a) are orthonormal for any a ∈ I2,
a 6= k
, k = 0, 1, . . . 2s − 1. Thus the system of functions ψ(s)(x+ a), a ∈ I2 is
orthonormal if and only if the system of functions, consisting of the function
(5.4) and its shifts, i.e.,
= −α2s−rψ(0)(x)−α2s−r+1ψ(0)
−· · ·−α2s−1ψ(0)
r − 1
(5.6) + α0ψ
+ · · ·+ α2s−r−1ψ(0)
2s − 1
r = 0, 1, . . . , 2s − 1 is orthonormal.
Set Ξ(0) = {ψ(0)(· + k
) : k = 0, 1, . . . , 2s − 1}T , Ξ(s) = {ψ(s)(· + k
) : k =
0, 1, . . . , 2s − 1}T . In view of (5.4), (5.6), Ξ(s) = DΞ(0), where
(5.7) D =

α0 α1 α2 . . . α2s−2 α2s−1
−α2s−1 α0 α1 . . . α2s−3 α2s−2
−α2s−2 −α2s−1 α0 . . . α2s−4 α2s−3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
−α2 −α3 −α4 . . . α0 α1
−α1 −α2 −α3 . . . −α2s−1 α0

Thus the system Ξ(s) is orthonormal if and only if the matrix D is unitary.
16 V. M. SHELKOVICH AND M. SKOPINA
Let u = (α0, α1, . . . , α2s−1)
T be a vector and

0 0 . . . 0 0 −1
1 0 . . . 0 0 0
0 1 . . . 0 0 0
. . . . . . . . . . . . . . . . . . .
0 0 . . . 1 0 0
0 0 . . . 0 1 0

be a 2s × 2s matrix. It is easy to see that
Aru = (−α2s−r,−α2s−r+1, . . . ,−α2s−1, α0, α1, . . . , α2s−r−1)T ,
r = 1, 2, . . . , 2s − 1. Thus D =
u,Au, . . . , A2
. It is significant that
u = −u. Consequently, in order to describe all matrixes D (or in other
words, all vectors u), we should find all vectors u = (α0, α1, . . . , α2s−1)
T such
that the system {Aru : r = 0, 1, 2, . . . , 2s − 1} is orthonormal.
In view of the fact that the system ψ(0)(x+a), a ∈ I2 forms an orthonormal
basis in W0, it is easy to see that the vector u0 = (1, 0, . . . , 0, 0)
T is one of
mentioned above vectors u. That is the system composed of vectors u0 and
Aru0 = (δ0 r, δ1 r, . . . , δ2s−2 r, δ2s−1 r)
T , r = 1, 2, . . . , 2s−1, is orthonormal, where
δi r is the Kronecker symbol.
Let us prove that the vector u = (α0, α1, . . . , α2s−1)
T already mentioned
above such that Aru, r = 0, 1, 2, . . . , 2s − 1 is orthonormal, can be expressed
by the formula u = Bu0 if and only if B is a unitary matrix such that AB =
BA. Indeed, let u = Bu0, where B is a unitary matrix such that AB =
BA. Then Aru = BAru0, r = 0, 1, 2, . . . , 2
s − 1. Since the system Aru0,
r = 0, 1, 2, . . . , 2s − 1 is orthonormal and the matrix B is unitary, the vectors
Aru, r = 0, 1, 2, . . . , 2s − 1 are orthonormal. Conversely, if the system Aru,
r = 0, 1, 2, . . . , 2s−1 is orthonormal, taking into account that the system Aru0,
r = 0, 1, 2, . . . , 2s − 1 is orthonormal, we conclude that there exists a unitary
matrix B such that Aru = B(Aru0), r = 0, 1, 2, . . . , 2
s − 1. Since A2su = −u,
u0 = −u0, we have an additional relation A2
u = BA2
u0. It follows from
the above relations that (AB − BA)(Aru0) = 0, r = 0, 1, 2, . . . , 2s − 1. Since
the vectors Aru0, r = 0, 1, 2, . . . , 2
s − 1 form a basis in the 2s-dimensional
space, we conclude that AB = BA.
Thus we have D =
Bu0, BAu0, . . . , BA
2s−1u0
It is clear that the eigenvalues of A and the corresponding normalized eigen-
vectors are
(5.8) λr = −eiπ
and vr =
(vr)1, . . . , (vr)2s
, respectively, where
(5.9) (vr)l = 2
−s/2(−1)le−iπ
l, l = 0, 1, 2, . . . , 2s − 1,
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 17
r = 0, 1, 2, . . . , 2s − 1. As is well known, the matrix A can be represented as
A = CÃC−1, where
λ0 0 . . . 0
0 λ1 . . . 0
. . .
0 0 . . . λ2s−1
is a diagonal matrix, C =
v0, v1, . . . , v2s−1
. Since C is a unitary matrix, the
matrix B = CB̃C−1 is unitary if and only if B̃ is unitary. On the other hand,
AB = BA if and only if ÃB̃ = B̃Ã. Moreover, since according to (5.8) λk 6= λl,
whenever k 6= l, all unitary matrix B̃ such that ÃB̃ = B̃Ã, are given by
γ0 0 . . . 0
0 γ1 . . . 0
. . .
0 0 . . . γ2s−1
 ,
where γk ∈ C, |γk| = 1. Hence, all unitary matrix B such that AB = BA, are
given by B = CB̃C−1, where B̃ is the above diagonal matrix.
By using formula (5.9), one can calculate
αk = (Bu0)k = (CB̃C
−1u0)k =
2s−1∑
γr(vr)k(vr)0
= 2−s(−1)k
2s−1∑
−iπ 2r+1
k, k = 0, 1, 2, . . . , 2s − 1,
where γk ∈ C, |γk| = 1. Thus (5.5) holds.
Taking into account that Ξ(0) = D−1Ξ(s), we conclude that if we define
ψ(s)(x) by formula (5.4), where αk is given by (5.5), k = 0, 1, 2, . . . , 2
s − 1,
then the system of functions {ψ(s)(· − a) : a ∈ a ∈ I2} is orthonormal and
forms the orthonormal basis in W0.
Consequently, all dilatations and shifts of the function (5.4) form 2-adic
orthonormal wavelet basis in L2(Q2). �
It is clear that
γa (x) dx = 0, and in view of Lemma 3.1, ψ
γa (x) belongs
to the Lizorkin space ∈ Φ(Qn2 ).
5.2. Real wavelets. Using formulas (5.5), one can extract all real wavelet
functions (5.4).
Let s = 1. According to (5.2), (5.3),
(5.10) ψ(1)(x) = cos θ ψ(0)
+ sin θ ψ(0)
is the real wavelet function.
18 V. M. SHELKOVICH AND M. SKOPINA
Let s = 2. Set γr = e
iθr , r = 0, 1, 2, . . . , 2s − 1. Then (5.5) imply that the
wavelet function ψ(1)(x) is real if and only if
sin θ1 + sin θ2 + sin θ3 + sin θ4 = 0,
cos θ1 − cos θ2 + cos θ3 − cos θ4 = 0,
sin θ1 − sin θ2 − sin θ3 + sin θ4 =
cos θ1 + cos θ2 − cos θ3 − cos θ4,
sin θ1 − sin θ2 − sin θ3 + sin θ4 =
−(cos θ1 + cos θ2 − cos θ3 − cos θ4).
The last relations are equivalent to the system
sin θ1 = − sin θ4, cos θ1 = cos θ4,
sin θ2 = − sin θ3, cos θ2 = cos θ3.
Thus for s = 2 the real wavelet functions (5.4) is represented as
ψ(1)(x) =
(cos θ1 + cos θ2)ψ
(cos θ1 − cos θ2 + sin θ1 + sin θ2)ψ(0)
(sin θ1 − sin θ2)ψ(0)
(5.11) +
(cos θ1 − cos θ2 − sin θ1 − sin θ2)ψ(0)
In particular, for the special cases θ1 = θ2 = θ, θ1 = −θ2 = θ, θ1 = θ2+π2 = θ,
we obtain one-parameter families of the real wavelet functions
(5.12)
ψ(1)(x) = cos θψ(0)
+ sin θψ(0)
ψ(1)(x) = cos θψ(0)
sin θψ(0)
sin θψ(0)
ψ(1)(x) = 1
(cos θ − sin θ)ψ(0)
(cos θ + sin θ)ψ(0)
(cos θ − sin θ)ψ(0)
respectively.
Acknowledgments
The authors are greatly indebted to E. Yu. Panov for fruitful discussions.
p-ADIC HAAR MULTIRESOLUTION ANALYSIS 19
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Department of Mathematics, St.-Petersburg State Architecture and Civil
Engineering University, 2 Krasnoarmeiskaya 4, 190005, St. Petersburg, Rus-
sia. Phone: +7 (812) 2517549 Fax: +7 (812) 3165872
E-mail address : shelkv@vs1567.spb.edu
Department of Applied Mathematics and Control Processes, St. Peters-
burg State University, Universitetskii pr.-35, Petrodvorets, 198504 St. Pe-
tersburg, Russia. Phone: +7 (812) 51326090 Fax: +7 (812)
E-mail address : skopina@ms1167.spb.edu
	1. Introduction
	1.1. p-Adic wavelets and pseudo-differential operators.
	1.2. Contents of the paper.
	2. p-Adic distributions
	3. The p-adic Lizorkin spaces
	4. Construction of multiresolution analysis
	4.1. p-Adic multiresolution analysis.
	4.2. p-Adic refinement equation.
	4.3. Construction of 2-adic Haar multiresolution analysis.
	5. Description of 2-adic Haar bases
	5.1. Complex wavelets.
	5.2. Real wavelets.
	Acknowledgments
	References
