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ANNALES MATHÉMATIQUES
BLAISE PASCAL
Cui Minggen, Zhang Yanying
The Heisenberg uncertainty relation in harmonic
analysis on p-adic numbers field
Volume 12, no1 (2005), p. 181-193.
<http://ambp.cedram.org/item?id=AMBP_2005__12_1_181_0>
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Annales Mathematiques Blaise Pascal 12, 181-193 (2005)
The Heisenberg uncertainty relation in
harmonic analysis on p-adic numbers field
Cui Minggen
Zhang Yanying
Abstract
In this paper, two important geometric concepts– grapical cen-
ter and width, are introduced in p-adic numbers field. Based on the
concept of width, we give the Heisenberg uncertainty relation on har-
monic analysis in p-adic numbers field, that is the relationship between
the width of a complex-valued function and the width of its Fourier
transform on p-adic numbers field.
1 Introduction
In reference [1], wavelet transform is introduced to the field of p-adic numbers.
In references [2] and [5], some theory of wavelet analysis and affine frame on
harmonic analysis are introduced to the field of p-adic numbers respectively
on the basis of a mapping P: R+ ∪ {0} → Qp(field of p-adic numbers)
In this paper, based on the same mapping P we will give the Heisenberg
uncertainty relation in harmonic analysis on p-adic numbers field as
4f4 bf ≥ 14π2
where ∆f , ∆ bf are the widths of function f and its transform f̂ respectively.
The field of the p-adic numbers is defined as the completion of field Q of
rationals with respect to the p-adic metric induced by the p-adic norm |·|p(see
[6]). A p-adic numbers xp 6= 0 is uniquely represented in the canonical form
xp = p
−r
∞∑
k=0
xkp
k, |xp|p = pr, (1.1)
where r∈Z and xk∈Z such that 0 ≤ xk ≤ p− 1, x0 6= 0, For xp, yp ∈ Qp, we
define xp < yp either when |xp|p < |yp|p or when |xp|p = |yp|p, and there exist
181
M.G.Cui, Y.Y.Zhang
an integer j such that x0 = y0, · · · , xj−1 = yj−1, xj < yj from the viewpoint of
(1.1). By interval [ap, bp], we mean the set defined by {xp ∈ Qp|ap≤xp≤bp}.
The mapping P: R+ ∪ {0} → Qp are introduced in the references [5] and
[2] as
P(0) = 0; P
(
pr
∞∑
k=0
xkp
−k
)
= p−r
∞∑
k=0
xkp
k ∈ Qp . (1.2)
It is known that if x
R
= pr
∑n
k=0 xkp
−k ∈ R+∪{0}, x0 6= 0 and 0 ≤ xk ≤ p−1,
then it has another expression
x
R
= pr
n−1∑
k=0
xkp
−k + (xn − 1)p−n + (p− 1)
∞∑
k=n+1
p−k. (1.3)
that we won’t use it in this paper. Let M
R
be the set of numbers that
is expressed by formula (1.3) and Mp = P(MR). Let Br(ap) = {xp∈Qp ‖
|xp − ap|p≤pr, r∈Z}, Sr(ap) = {xp∈Qp ‖ |xp − ap|p = pr, r∈Z}. According to
(1.2), we reach a conclusion that for an interval [a
R
, b
R
] in R+ ∪ {0} and its
corresponding interval [ap, bp] in Qp
P{Br(ap)} = [0, pr+1), (1.4)
P{Sr(ap)} = [pr, pr+1), (1.5)
P{[ap, bp)} = [aR , bR), (1.6)
|a
R
− b
R
|≤p|aP − bp|p, (1.7)
where P(a
R
) = ap,P(bR) = bp(see [3]). Let f be a complex-valued function
on Qp, for xp ∈ Qp \Mp, let
f(xp) = f(P ◦P−1(xp)) = (f ◦P)(xR)
def
= f
R
(x
R
), (f
R
= f ◦P), x
R
= P−1xp .
(1.8)
From (1.7), we know that the inverse mapping P−1 is continuous on
Qp\Mp
2 A Haar measure on Q and integration
In this section, a Haar measure is constructed by using the mapping P of
R+ ∪ {0} into Qp\Mp and the Lebesque measure on R+ ∪ {0}. The symbol
182
The Heisenberg uncertainty relation
∑
is the set of all compact subsets of Qp, and S is the σ − ring generated
by
∑
.
Definition 2.1: Let E∈S, and put Ep = E\Mp, and ER = P−1(Ep). If ER
is a measurable set on R+∪{0}, then we call E a measurable set on Qp, and
define a set function µp(E) on S:
µp(E) =
1
p
µ(E
R
) (2.1)
where µ(E
R
) is the Lebesque measure on E
R
. This µp(E) is called the mea-
sure on E.
By the Definition 2.1, some examples can be given immediately:
(1) Let ap, bp ∈ Qp, then µp{[ap, bp]} = (bR − aR)/p (see (1.7))
(2) µp{Br(0)} = pr (see (1.4))
(3) µp{Sr(0)} = pr(1− 1p) (see (1.5))
(4) Let {Bri(ai)}i be disjoint discs covering E, by the definition of measure
µp and definition of Lebesque exterior measure on R+∪{0}, it is evident that
µp(E) = inf
ri∈Z
µp{∪iBri(ai)} (2.2)
(5) µp(Mp) = 0.
It is obvious that µp, by Definition 2.1, is countably additive. In order to
prove that µp is a Haar measure, we will give the following lemma.
Lemma 2.2: If α ∈ Qp, then
µp{Br(α)} = µp{Br(0)}. (2.3)
Proof: 1◦ Let α = p−r1 , for r1 > r, r1, r∈Z, and put x = p−r1 +
p−r
∑
0≤xk<∞ xkp
k, x0 6=0, 0≤xk < p. Then E is the set of all these p-adic num-
bers when xk change for k = 0, 1, ..., p − 1. We write Ep = {xp|xp∈E\Mp}.
For xp∈Ep, let
P−1(xp) = p
r1 + pr
∑
0≤K<∞
xkp
−k (2.4)
then M
R
= [pr1 , pr1 +pr+1) is the set of all real numbers as presented in (2.4)
(see(1.4)). Hence
µp(α + Br(0)) = µp(Br(α)) =
1
p
µ(E
R
) = pr = µp(Br(0)) (2.5)
183
M.G.Cui, Y.Y.Zhang
2◦ Let α = p−r2 , for r2≤r, r2, r∈Z. then α + Br(0) = Br(α) = Br(0), by
α∈Br(0). So that
µp(Br(α)) = µp(Br(0)) (2.6)
3◦ Let α = p−r3
∑
0≤k<∞ αkp
k, and put αn = p−r3
∑
0≤k≤n αkp
k, applying
to the result of 1◦ and 2◦ repeatedly in this case, we have
µp(α
n + Br(0)) = µp(Br(α
n)) = µp(Br(0)) (2.7)
However
lim
n→∞
µp(Br(α
n)) = lim
n→∞
1
p
µ{P−1(Brp(αn))} (2.8)
where Brp(αn) = Br(αn)\Mp. By the continuity of the mapping P−1
(see(1.7)), we obtain
lim
n→∞
1
p
µ{P−1(Brp(αn))} =
1
p
{P−1µ{P−1(Brp(α))} = µp{Br(α)} (2.9)
The part 3◦ follows from (2.7), (2.8) and (2.9).
Theorem 2.3: (The translation invariance of the measure µp) Let E∈S and
let α∈Qp, then
µp(α + E) = µp(E) (2.10)
Proof: Let {Bri(ai)}∞i=1 be disjoint discs covering E, then {Bri(ai +α)}∞i=1
are disjoint discs covering α + E. By the formula (2.2) in the example 4, we
have
µp(α + E) = inf
ri∈Z
µp{∪{α + Bri(ai)} (2.11)
Applying the lemma 2.2 to the right side of the above formula, then
inf
ri∈Z
µp{∪iBri(α + ai)}
= inf
ri∈Z
∑
i
µp{Bri(α + ai)}
= inf
ri∈Z
∑
i
µp{Bri(ai)}
= inf
ri∈Z
µp{∪iBri(ai)}
= µp(E)
184
The Heisenberg uncertainty relation
Therefore, µp is a Haar measure.
According to the above definition of Haar measure, we can define the
integration over measurable sets E in Qp (firstly define the integration of
simple functions, then regard the limit of integration of simple functions as
the definition of the integration of general functions (see [4]))∫
E
f(xp)dµp (2.12)
By the theorem 2.3, the definition of measure and (1.8), we have the
following theorem
Theorem 2.4: Suppose f(xp) is a complex-valued function on Qp, then f(xp)
is integrable over the interval [ap, bp] (ap, bp ∈ Qp), if and only if the real
function f
R
(x
R
) defined on R+ ∪ {0} is integrable over the interval [a
R
, b
R
],
and ∫
[ap,bp]
f(xp)dµ(xp) =
1
p
∫ b
R
a
R
f(x
R
)dµ(x
R
) (2.13)
where f
R
(x
R
) is defined by (1.8), and P(x
R
) = xp,P(aR) = ap,P(bR) =
bp, ap, bp∈̄Mp
Corollary 2.5: If f(xp) is a bounded continuous function on the interval
[ap, bp]⊂Qp, then f(xp) is integrable over [ap, bp], where [ap, bp] can be Qp.
Notice that under the condition of theorem f
R
(x
R
) is a bounded piecewise
continuous function on R+ ∪ {0} by (1.4), By the theorem 2.4, f(xp) is
integrable.
3 The indefinite integral and derivative of
complex-valued function in Qp
Definition 3.1: Let f be a complex-valued function defined in Qp and for
∀xp ∈ Qp, f is integrable on interval [ap, bp], then
f(xp) =
∫ xp
0
gdxp (3.1)
is called on indefinite integral of g.
185
M.G.Cui, Y.Y.Zhang
Definition 3.2: Let f be a complex-valued function defined in Qp, if there
exist an integrable complex-valued function g such that
f(xp) =
∫ xp
0
gdxp, xp ∈ Qp (3.2)
then g(xp) is called the derivative of f , which we will denote as f ′(xp).
In formula (3.2), let f = 1 then
µ([0, xp]) =
∫ xp
0
dµ (3.3)
The equation (3.3) follows that
µ̄′(x)
def
= µ′([0, xp]) = 1 (3.4)
Theorem 3.3: For complex-valued functions f, h on Qp, if (f)R(xR) and
(h)
R
(x
R
) are absolutely continuous, then
f ′
R
(x
R
) = (f ′)
R
(x
R
)
(f(xp)h(xp))
′ = f ′(xp)h(xp) + f(xp)h
′(xp) (3.5)
Proof: Let f ′ = g and g(xp) = g(P(xR)) = (g ◦ P)(xR) = gR(xR). By
definition (3.2) and theorem 2.4, we have
f(xp) =
∫ xp
0
g(xp)dxp
=
∫ x
R
0
g
R
(x
R
)dx
R
= f
R
(x
R
)
and therefore
(f
R
)′(x
R
) = g
R
(x
R
) = g(xp) = f
′(xp) = (f
′)
R
(x
R
) (3.6)
and
(f(xp)h(xp))
′ = (f
R
(x
R
)h
R
(x
R
))′
= (f
R
)′(x
R
)h
R
(x
R
) + f
R
(x
R
)(h
R
)′(x
R
)
= (f ′)
R
(x
R
)h
R
(x
R
) + f
R
(x
R
)(h′)
R
(x
R
)
= f ′(xp)h(xp) + f(xp)h
′(xp)
186
The Heisenberg uncertainty relation
From (3.6) it follows that
Corollary 3.4: If a complex-valued function h(x
R
) is absolutely continuous
on R+ ∪ {0}, then f(xp)
def
= (hP−1)(xp) is derivable on Qp \Mp.
Corollary 3.5: A locally constant function is derivable on Qp \Mp, and its
derivative is equal to 0.
Similarly, we can prove
Theorem 3.6: If f is derivable on [ap, bp],then∫ bp
ap
f ′(xp)dµ = f(bp)− f(ap) (3.7)
4 Center and width of the graph of f
In this section, we will introduce the concepts of center and width of complex-
valued function graph in the filed of p-adic numbers Qp.
Definition 4.1: Let f be a complex-valued function of p-adic variable. We
define the center t
f
of the graph {(xp, f(xp))|xp ∈ Qp} by
t(R)
f
def
=
∫
Qp\Mp
P−1(xp)|f(xp)|2dxp/
∫
Qp\Mp
|f(xp)|2dxp
t
f
= P(t(R)
f
)
 (4.1)
if the integral (4.1) exists.
Definition 4.2: For a complex-valued function of p-adic variable,we define
the width of f by
4f =
(∫
Qp
|xp − tf |2|f(xp)|2dxp/
∫
Qp
|f(xp)|2dxp
)1/2
(4.2)
if the integral (4.2) exists.
187
M.G.Cui, Y.Y.Zhang
Theorem 4.3: Let P(t(R)f −aR) = P(t
(R)
f )−P(aR), aR = P−1(a), a ∈ Qp\Mp.
(1) If f is increasing, then t
Taf
= t
f
− a
(2) Suppose suppf⊂Br(0). For a = p−β, if β > r, then tTaf = tf − a
(3) For a = p−β, β ∈ Z, then
tsaf = atf ,
where Taf(xp) = f(xp + a), Saf(xp) = f(
xp
a
).
Proof: (1) Under the condition of (1) in this theorem, using
P(x
R
+ a
R
) ≥ P(x
R
) + P(a
R
)
we have
(f ◦P)(x
R
+ a
R
)≥f(x + a)
where x
R
= P−1(xp), xp ∈ Qp \Mp. Hence
t(R)
Taf
=
∫
Qp\Mp
P−1(xp)|Taf(xp)|2dxp/
∫
Qp
|Taf(xp)|2dxp
=
∫
Qp\Mp
P−1(xp)|f(x + a)|2dxp/
∫
Qp
|f(xp)|2dxp
≤
∫
R+
x
R
|(f ◦P)(x
R
+ a
R
)|2dx
R
/
∫
Qp
|f(xp)|2dxp
=
∫
R+
(x
R
− a
R
)|(f ◦P)(x
R
)|2dx
R
/
∫
Qp
|f(xp)|2dxp
= −a
R
+
∫
R+
x
R
|(f ◦P)(x
R
)|2dx
R
/
∫
Qp
|f(xp)|2dxp
= −a
R
+
∫
Qp\Mp
P−1(xp)|f(xp)|2dxp/
∫
Qp
|f(xp)|2dxp
= −a
R
+ t(R)
f
(4.3)
where we used µ(Mp) = 0, and for xp, a∈Br(0) ∩ (Qp\Mp), xp + a∈Br(0).
On the other hand, using inequation P−1(x− a) ≥ P−1(x)−P−1(a), we can
easily obtain
t(R)
Taf
≥t(R)
f
− a
R
(4.4)
From (4.3) and (4.4), we have
t(R)
Taf
= t(R)
f
− a
R
.
188
The Heisenberg uncertainty relation
Finally, from the condition of (1) in theorem, we have
t
Taf
= P(t(R)
f
− a
R
) = P(t(R)
f
)−P(a
R
) = t
f
− a
.
Conclusion of (1) in theorem is proved. (2) and (3) can be proved simi-
larly.
Theorem 4.4: (1) If f(x) and a, tf satisfy the condition of theorem 3.3,
then
4Taf = 4f
(2) 4Saf = |a|p4f
Proof: For (1), we have
4Taf =
(∫
Qp
|xp − tTaf |
2
p|Taf |2(xp)dxp/
∫
Qp
|Taf |2(xp)dxp
)1/2
=
(∫
Qp
|xp − (tf − a)|2p|f(xp + a)|2dxp/
∫
Qp
|f(xp + a)|2dxp
)1/2
=
(∫
Qp
|tp − tf |2p|f(tp)|2dtp/
∫
Qp
|f(tp)|2dtp
)1/2
= 4f
(2) can be proved similarly.
After doing the preparation of section 1-4, we will give a theorem on har-
monic analysis which is about the relation of the width of complex function
in Qp and the width of its Fourier transform. This theorem is similar to the
Heisenberg uncertainty relation in quantum mechanics.
5 Main theorem
Lemma 5.1: Let xp ∈ Qp, then µ([0, xp]) ≤ |xp|p
189
M.G.Cui, Y.Y.Zhang
Proof: For xp ∈ Pp \Mp
xp = p
−r
∞∑
k=0
xkp
k ∈ Qp, x0 6= 0, 0≤xk ≤ p− 1
and therefore, we have
P−1(xp) = p
r−1
∞∑
k=0
xkp
−k≤pr−1(p− 1)
∞∑
k=0
p−k = |xp|p (5.1)
By definition of measure µp, we have
1
p
P−1(xp) = µ([0, xp])
which leads to
µ(xp)
def
= µ([0, xp]) ≤ |xp|p/p (5.2)
Theorem 5.2: Let f be complex-valued function of p-adic variable. If
f∈L2(Qp), f ′∈L2(Qp) and
lim
|bp|p→∞
|bp|p|f(bp)|2 = 0, f(0) = 0 (5.3)
then the following inequality is valid:
1
4π
≤ 4f4 bf (5.4)
where f̂ is the transform of f ,
f̂(ξp) =
∫
Qp
f(xp)exp(2πi{ξpxp})dxp
and by means of representation (1.1), {xp} is defined as
{xp} =
{
0 if r(xp) ≥ 0 or xp = 0
pr(x0 + x1p + · · ·+ x|r|−1p|r|−1) if r(xp) < 0
Inequality (5.4) is called the Heisenberg uncertainty relation in harmonic
analysis on p-adic numbers field.
190
The Heisenberg uncertainty relation
Proof: By using (3.4) and theorem 3.3, we have(
µ̄(xp − tf )|f(xp)|2
)′
=
(
µ̄(xp − tf )f(xp)χp(t bf xp)f(xp)χp(t bf xp)
)′
= |f(xp)|2 + µ̄(xp − tf ) (f(xp)χp(t bf xp)
)′
f(xp)χp(t bf xp)
+µ̄(xp − tf )f(x)χp(t bf xp)[f(xp)χp(t bf xp)]′ (5.5)
Therefore, from (3.7) we have∫ bp
0
|f(xp)|2dx = µ̄(xp − tf )|f(xp)|2|
bp
0
−
∫ bp
0
µ̄(xp − tf )f(xp)χp(t bf xp)
(
f(xp)χp(t bf xp)
)′
dxp
−
∫ bp
0
µ̄(xp − tf )f(xp)χp(t bf xp)[f(xp)χp(t bf xp)]′dxp (5.6)
where the function χp(t bf xp) = exp(2πi{t bf xp})
By taking the limit of (5.5) as |b|p →∞ and using (5.2),(5.3), we obtain∫
Qp
|f(xp)|2dxp
≤ 2
(∫
Qp
(µ̄(xp − tf ))2|f(xp)|2dxp
)1/2(∫
Qp
|[f(xp)χp(t bf xp)]′|2dxp
)1/2
=
(∫
Qp
(µ̄(xp − tf ))2|f(xp)|2dxp
)1/2(∫
Qp
|[f(·)χp(t bf ·)]′
∧
(ξ)|2dξ
)1/2
= 2
(∫
Qp
(
µ̄(xp − tf )
)2 |f(xp)|2dxp)1/2(∫
Qp
4π2|ξ|2|f̂(ξ + t bf )|2dξ
)1/2
(5.7)
where we used the Hölder inequality for the integral and f ′(·)∧(ξ) = −2πiξf̂(ξ),
(f, f)L2(Qp) = (f̂ , f̂)L2(Qp), (f(·)χp(a·))
∧ (ξ) = f̂(ξ + a) From (5.2), we have
1
4π
≤
(∫
Qp
|xp − tf |2p|f(xp)|2dxp/
∫
Qp
|f(xp)|2dxp
)1/2
·
191
M.G.Cui, Y.Y.Zhang
(∫
Qp
|ξp − t bf |2p|f̂(ξp)|2dξp/
∫
Qp
|f̂(ξp)|2dξp
)1/2
= 4f4 bf (5.8)
Hence we have completed our proof.
References
[1] M. G. Cui. On the wavelet transform in the field Qp of p-adic numbers.
Appl. Comput. Harmonic Analysis, 13:162–168, 2002.
[2] M. G. Cui, H.M.Yao, and H.P.Liu. The affine frame in p-adic analysis.
Annales Mathematiques Blaise Pascal, 10:297–303, 2003.
[3] G.H.Gao and M.G.Cui. Calculus on the field Qp of p-adic numbers. J. of
Natural Science of Heilongjiang University, 3:15–16, 2003.
[4] Paul R. Halmos. Measure Theory. Beĳing Scientific Publishing
House(Chinese Translation), Beĳing, 1965.
[5] S. V. Kozyrev. Wavelet theory as p-adic apectral analysis, izv. russ. Akad.
Nauk, Ser. Math., 66:149–158(Russian), 2002.
[6] V.S.Vladimirov, I.V.Volovich, and E.I.Zelenov. p-adic analysis and math-
ematical physics. Internat. Math. Res. Notices, 13:6613–663, 2996.
192
The Heisenberg uncertainty relation
Cui Minggen
Harbin Institute of Technology
Department of Mathematics
No.2 WenHua west road
WeiHai, ShanDong, 264209
CHINA
cmgyfs@263.net
Zhang Yanying
Harbin Normal University
Department of Information Science
HeXing Road
Harbin,HeiLongJiang, 150080
CHINA
193


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Problème de Dirichlet pour les fonctions $\alpha $-harmoniques sur les domaines coniques

Khodr Shamseddine

Martin Berz

Analytical properties of power series on Levi-Civita fields

George Csordas

Alain Escassut

The Laguerre inequality and the distribution of zeros of entire functions

Minggen Cui

Xueqin Lv

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Minggen Cui, Xueqin Lv
Weyl-Heisenberg frame in p-adic analysis
Volume 12, no1 (2005), p. 195-203.
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Annales Mathematiques Blaise Pascal 12, 195-203 (2005)
Weyl-Heisenberg frame in p-adic analysis
Minggen Cui
Xueqin Lv
Abstract
In this paper, we establish an one-to-one mapping between complex-
valued functions defined on R+ ∪ {0} and complex-valued functions
defined on p-adic number field Qp, and introduce the definition and
method of Weyl-Heisenberg frame on hormonic analysis to p-adic anyl-
ysis .
1 Introduction
Wavelet transform was introduced to the field of p-adic numbers in [1]. In
[4],[3] some theory of wavelet analysis and affine frame introduced to the field
of p-adic numbers,respectively,on the basis of a mapping P : R+ ∪ {0} →
Qp(field of p−adic numbers). This paper considers on the basis of the
mapping P , gives the Weyl-Heisenberg frame in field of p-adic numbers.
The field Qp of the p-adic numbers is defined as the completion of field
Q of rational with respect to the p-adic metric induced by the p-adic norm
|.|p,(see [5]). A p-adic number x 6= 0 is uniquely represented in the canonical
form
x = p−r
∞∑
k=0
xkp
k, |x|p = pr (1.1)
where p is prime and r ∈ Z (Z is integer set) , 0 ≤ xk ≤p−1, x0 6= 0. For
x, y ∈ Qp, we define x < y,either when |x|p < |y|p or when |x|p = |y|p, but
there exists an integer j such that x0 = y0, · · · , xj−1 = yj−1, xj < yj from
viepoint of (1.1).By interval [a, b], we mean the set defined by {x ∈ Qp|a ≤
x ≤ b}.
It is known that if x = pr
∑n
k=0 xkp
−k ∈ R+ ∪ {0}and x0 6= 0, 0 ≤ xk ≤
p− 1, k = 1, 2, · · · , then there is another expression;
x = pr(
n−1∑
k=0
xkp
−k + (xn − 1)p−n + (p− 1)
∞∑
k=n+1
p−k) (1.2)
195
M. G. Cui and L. Q. Lv
But we won’t use that expression (1.2) in this paper.
A mapping P : R+ ∪ {0} → Qp was introduced in [4],[3],[2], as for
x = pr
∑∞
x=0 xkp
−k ∈ R, x0 6= 0, 0 ≤ xk ≤ p− 1, k = 1, 2, · · ·
P (x) = p−r
∞∑
k=0
xkp
k
Let Mp = P (MR),
M
R
= {x
R
|x
R
= pr
n−1∑
k=0
xkp
−k + (n− 1)p−n + (p− 1)
∞∑
k=0
p−k, n ∈ Z+ ∪ {0}}
In the following to distinguish between real number field R and p-adic
number field Qp, number in R denotes by the subscript R, and number with
the subscript p belongs to Qp. For example xR , aR , bR in R; xp , ap in Qp.
In [2] a measure is constructed using the mapping P from R+ ∪ {0} into
Qp\Mp and Lebesgue measure on R+ ∪ {0}, the symbol
∑
is the set of all
compact subsets of Qp, and S is the σ − ring generated by
∑
.
Definition 1.1:[2] Let E ∈ S, and put EP = E\MP , and ER = P−1(EP ).If
ER is a measurable set on R+∪{0}, the we call E is a measurable set on Qp,
and define a set function µp(E) on S
µp(E) =
1
p
µ(ER)
where µ(ER) is the Lebesgue measure on ER.This µp(E) is called the
measure on E.
By the Definition 1.1, some examples can given immediately:
(1) Let ap, bp ∈ QP , then µp{[ap, bp]} = (bR − aR)/p
(2) Let Br(0) = {xp||xp|p ≤ pr, xp ∈ Qp}, then µp{Br(0)} = pr
(3) Let Sr(0) = {xp||xp|p = pr, xp ∈ Qp}, then µp{Sr(0)} = pr(1− 1P )
(4) µp{Mp} = 0
According to the above definition 1.1 of measure, we can define integra-
tion over measurable sets E in Qp∫
E
f(xp)dµp(xp) or
∫
E
f(xp)dxp
196
Weyl-Heisenberg frame in p-adic analysis
By the definition 1.1 of measure we have following theorem.
Theorem 1.2:(see [2]) Suppose f(xp) is a Complex-Valued function on Qp,
the f(xp) is integrable over the interval [ap, bp](ap, bp ∈ Qp), if and if the real
function fR(xR) defined on R+ ∪ {0} is Lebesgue integrable over the interval
[a
R
, b
R
], and ∫
[ap,bp]
f(xp)dxp =
1
p
∫ b
R
a
R
f(x
R
)dx
R
(1.3)
where f
R
(x
R
) is defined
f(xp) = f(P ◦ P−1(xp)) = (f ◦ P )(xR)
def
= f
R
(x
R
), xp = P (xR) ∈ Qp\Mp
2 Weyl-Heisenberg frame on p-adic number
field
In real analysis, if there exists constants A and B, A, B > 0, such that
A‖f‖2L2(R) ≤
∑
m,n
|(f, gm,n)L2(R)|2 ≤ B‖f‖2L2(R)
holds for ∀f ∈ L2(R), then gmn(x) is called the Weyl-Heisenberg frame, where
g ∈ L2(R),p0, q0 ∈ R,gmn(x) = g(x−nq0)e2πimp0x, m, n ∈ Z and (f, gmn)L2(R)
is inner product in L2(R),
(f, gmn)L2(R) =
∫
R
f(x)gmn(x)dx.
In this section we give the definition of a Weyl-Heisenberg frame in Qp by
gmn(xp) = g(αmn(xp)− xp) exp(2πimp0ρ(xp)) (2.1)
where
αmn(xp) = P (|xR + nq0|) + xp (2.2)
and m, n, p0, q0 ∈ Z, xR = P−1(xp),xp ∈ Qp\Mp . If gmn(xp) satisfies the
frame condition :
A‖f‖2L2 ≤
∑
m,n
|(f, gm,n)L2|2 ≤ B‖f‖2L2 , A,B > 0,∀f ∈ L2(Qp)
197
M. G. Cui and L. Q. Lv
then we have
f(xp) =
∑
m,n
(f, g∗m,n)L2gm,n(xp) =
∑
m,n
(f, gm,n)g
∗
m,n(xp), xp ∈ Qp
where {g∗m,n} is the dual frame of {gm,n}:
g∗m,n = S
−1gm,n
and S is the frame operator:
Sf =
∑
m,n
(f, gm,n)L2gm,n
and
(f, gmn)L2 =
∫
Qp
f(xp)gmn(x)dx
Theorem 2.1: Suppose f, g ∈ L2(Qp) are complex-valued functions defined
on Qp , p0 = prp , q0 = prq , rp, rq ∈ Z, if support ĝR(|w|) ⊂ [−12q0 ,
1
2q0
], and
∃A, B > 0, such that
A ≤
∑
m∈Z
|ĝ
R
(|w −mp0|)|2 ≤ B, w ∈ R ,∀w 6= 0,
then functions gmn(xp) defined by (2.1) construct a frame of L2(Qp), where
g
R
(|t|) = (g ◦ P )(|t|) = g(xp), (P (|t|) = xp, gR = g ◦ P ), t ∈ R, and ĝR(|w|) is
the Fourier transform of g
R
(|t|) in real analysis
ĝ
R
(|w|) =
∫
R
g
R
(|t|) exp(−2πiwt)dt
Proof. From formula (2.2) or g(αmn(xp)− xp) = (g ◦ P )(|xR + nq0|) =
g
R
(|x
R
+ nq0|), for ∀f ∈ L2(Qp), we have∑
m,n∈Z
|(f, gm,n)L2|2 =
∑
m,n∈Z
|
∫
Qp
f(xp)gm,n(xp)dxp|2
=
∑
m,n∈Z
|
∫
Qp
f(xp)g(α
(n)
m (xp)− xp) exp(−2πimp0ρ(xp))|2
198
Weyl-Heisenberg frame in p-adic analysis
=
1
p
∑
m,n∈Z
|
∫
R+
f
R
(x
R
)g
R
(|x
R
+ nq0|) exp(−2πimp0xR)dxR |2 (2.3)
where we used (1.3) and f
R
(x
R
) = (f ◦ P−1)(xp) in section 1
Let
f+
R
(x) =
{
f
R
(x), x ≥ 0
0, x < 0
From (2.3), we obtain ∑
m,n∈Z
|(f, gm,n)|2
=
∑
m,n∈Z
|
∫
R
f+
R
(x)(g
R
(|x + nq0|) exp(−2πimp0x)dx |2
=
∑
m,n∈Z
|
∫
R
f̂+
R
(w)[g
R
(| ·+nq0|) exp(2πimp0·)]∧(w)dw|2 (2.4)
where sign “·” is the argument on the function, for Fourier transform. But
[gR(| ·+nq0|) exp(2πimp0·)]∧(w)
= ĝ
R
(|w −mp0|) exp(2πinq0(w −mp0))
Hence from the support ĝ ⊂ [− 1
2q0
, 1
2q0
] in condition of the theorem and
(2.4) we have ∑
m,n∈Z
|(f, gmn)|2
=
∑
m,n∈Z
|
∫
R
f̂+
R
(w + mp0)ĝR(|w|) exp(−2πinq0w)dw|2
=
∑
m,n∈Z
|
∫ 1
2q0
− 1
2q0
f̂+R (w + mp0)ĝR(|w|) exp(−2πinq0w)dw|
2 (2.5)
199
M. G. Cui and L. Q. Lv
We know that
cn = q0
∫ 1
2q0
−1
2q0
f̂+
R
(w + mp0)ĝR(|w|) exp(−2πinq0w)dw, n ∈ Z
are Fourier coefficient of f̂
R
(w + mp0)ĝR(|w|) on [−12q0 ,
1
2q0
]. Hence by virtue
of Parseval equality we have∑
n∈Z |cn|2
=
∞∑
n=−∞
|q0
∫ 1
2q0
−1
2q0
f̂+
R
(w −mp0)ĝR(|w|) exp(−2πinq0w)dw|2
= q0
∫ 1
2q0
− 1
2q0
|f̂+
R
(w + mp0)ĝR(|w|)|2dw
= q0
∫
R
|f̂+R (w + mp0)ĝR(|w|)|
2dw
= q0
∫
R
|f̂+
R
(w)ĝ
R
(|w −mp0|)|2dw (2.6)
Comparing (2.5) and (2.6), we have∑
m,n∈Z
| (fp, gm,n)L2 |2=
1
q0
∫
R
|f̂+
R
(w)|2G(w)dw (2.7)
where
G(w) =
∑
m∈Z
|ĝ
R
(|w −mp0|)|2
Finally, by virtue of the conditions of the theorem, we have
∑
m∈Z,n∈Z
|(f, gm,n)L2|2 =
{
≥ A
q0
∫
R
|f̂+
R
(w)|2dw
≤ B
q0
∫
R
|f̂+
R
(w)|2dw
But∫
R
|f̂+
R
(w)|2dw=
∫
R
|f+
R
(x)|2dx=
∫
R+
|f
R
(x
R
)|2dx
R
=
∫
Qp
|f(xp)|2dxp =‖f‖L2 .
Hence we completed our proof.
200
Weyl-Heisenberg frame in p-adic analysis
3 Dual frame
In the section, we will give a formula to calculate the dual frame. By (2.7),
we have
(Sf, f)L2 =
∑
m,n∈Z
|(f, gm,n)L2|2 =
1
q0
∫
R
f̂+
R
(w)G(w)f̂+
R
(w)dw
=
1
q0
∫
R
(f̂+
R
(·)G(·))∨(x)f+R (x)dx
=
1
q0
∫
R+
(f̂+
R
(·)G(·))∨(x
R
)f
R
(x
R
)dx
R
where sign “ v " is the inverse Fourier transform. Therefore
(Sf, f)L2 =
1
q0
∫
Qp
(f̂+
R
(·)G(·))∨(P−1(xp))f(xp)dxp
=
1
q0
((f̂+
R
(·)G(·))∨(P−1(xp)), f(xp))L2
Since f is an arbitrary function in L2(Qp), we conclude that
(Sf)(xp) =
1
q0
(f̂+k (·)G(·))
∨(P−1(xp))
or for x ∈ R+
⋃
{0} we conclude that
(Sf)
R
(x
R
) =
1
q0
(f̂+
R
(·)G(·))∨(x
R
), x
R
≥ 0 (3.1)
where (Sf)
R
= (Sf)P−1
Bases on (3.1), we will extend the domain of (Sf)
R
(x
R
) from R+
⋃
{0}
onto R such that (Sf)
R
(t), t ∈ R is an even function on R. Taking Fourier
transform on both sides of (3.1), we have
((Sf)
R
)∧(w) =
1
q0
f̂+
R
(w)G(w) (3.2)
201
M. G. Cui and L. Q. Lv
After replace f with S−1f in formula (3.2), we have
f̂
R
(w) =
1
q0
{(S−1f)+
R
}∧(w)G(w)
Which leads to
{(S−1f)+
R
}∧(w) = q0 f̂R(w)
G(w)
(3.3)
Then we take Fourier inverse transformation on both sides of (3.3), we have
(S−1f)+
R
(x) = {q0 f̂R(·)
G(·)
}∨(x)
So, for x ≥ 0,
(S−1f)
R
(x
R
) = {q0f̂R(·)
G(·)
}∨(x
R
)
is valied or
(S−1)f(xp) = {
q0f̂R(·)
G(·)
}∨(P−1(xp)) (3.4)
Finally, let f(xp) = gmn(xp) in formula (3.4) , we obtain
g∗m,n(xp) = {
q0 f̂R(·)
G(·)
}∨(P−1(xp))
References
[1] MingGen Cui and GuangHong Gao. On the wavelet transform in the field
Qp of p-adic numbers. Applied and Computational Hormonic Analysis,
13:162–168, 2002.
[2] MingGen Cui and YanYing Zhang. The Heisenberg uncertainty relation
in harmonic analysis on p-adic numbers field. Dans ce fascicule des An-
nales Mathématiques Blaise Pascal.
202
Weyl-Heisenberg frame in p-adic analysis
[3] HuanMin Yao MingGen Cui and HuanPing Liu. The affine frame in p-
adic analysis. Annales Mathématiques Blaise Pascal, 10:297–303, 2003.
math.66.
[4] S.V.Kozyrev. Wavelet theory as p-adic spectral analysis.
Izv.Russ.Akad.Nauk,Ser, 2:149–158, 2002.
[5] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov. p-adic Analysis and
Mathematical Physics. World Scientific, 1994.
Minggen Cui
Harbin Institute of Technology
Department of Mathematics
WEN HUA XI ROAD
WEIHAI Shan Dong, 264209
P.R.China
cmgyfs@263.net
Xueqin Lv
Harbin Normal University
Department of Information Science
HE XING ROAD
Harbin HeiLongJiang, 150001
P.R.China
lvxueqin@163.net
203


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