The Heisenberg uncertainty relation in harmonic analysis on <span class="mathjax-formula">$p$</span>-adic numbers field

BLAISE PASCAL

Cui Minggen, Zhang Yanying

The Heisenberg uncertainty relation in harmonic analysis on p-adic numbers field

Volume 12, n^o1 (2005), p. 181-193.

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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 inp-adic numbers field, that is the relationship between the width of a complex-valued function and the width of its Fourier transform onp-adic numbers field.

1 Introduction

In reference [1], wavelet transform is introduced to the field ofp-adic numbers.

In references [2] and [5], some theory of wavelet analysis and affine frame on harmonic analysis are introduced to the field ofp-adic numbers respectively on the basis of a mappingP: R⁺∪ {0} →Q_p(field ofp-adic numbers)

In this paper, based on the same mappingP we will give the Heisenberg uncertainty relation in harmonic analysis onp-adic numbers field as

4_f4

fb≥ 1 4π² where∆_f, ∆

fbare the widths of functionf and its transformfbrespectively.

The field of thep-adic numbers is defined as the completion of field Qof rationals with respect to thep-adic metric induced by thep-adic norm|·|_p(see [6]). Ap-adic numbersx_p6= 0is uniquely represented in the canonical form

x_p=p^−r

∞

X

k=0

x_kp^k,|x_p|_p=p^r, (1.1)

wherer∈Z and x_k∈Z such that0≤x_k ≤p−1, x₀ 6= 0, For x_p, y_p∈Q_p, we definex_p< y_peither when|x_p|_p<|y_p|_por when|x_p|_p=|y_p|_p, and there exist

an integerjsuch thatx₀=y₀,· · ·, xj−1=yj−1, x_j < y_jfrom the viewpoint of (1.1). By interval [a_p, b_p], we mean the set defined by {x_p∈Q_p|a_p≤x_p≤b_p}.

The mappingP: R⁺∪ {0} →Q_pare introduced in the references [5] and [2] as

P(0) = 0; P p^r

∞

X

k=0

x_kp^−k

!

=p^−r

∞

X

k=0

x_kp^k∈Q_p . (1.2)

It is known that ifx_R =p^rPn

k=0x_kp^−k∈R⁺∪{0}, x₀6= 0and0≤x_k ≤p−1, then it has another expression

x_R =p^r

n−1

X

k=0

x_kp^−k+ (x_n−1)p⁻ⁿ+ (p−1)

∞

X

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 M_p = P(MR). Let B_r(a_p) = {x_p∈Q_p k

|x_p−a_p|_p≤p^r, r∈Z}, S_r(a_p) ={x_p∈Q_pk |x_p−a_p|_p=p^r, 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 [a_p, b_p] inQ_p

P{B_r(a_p)}= [0, p^r+1), (1.4) P{S_r(a_p)}= [p^r, p^r+1), (1.5) P{[a_p, b_p)}= [a_R, b_R), (1.6)

|a_R−b_R|≤p|a_P −b_p|_p, (1.7) where P(a_R) =a_p,P(b_R) =b_p(see [3]). Letf be a complex-valued function onQ_p, forx_p∈Q_p\M_p, let

f(x_p) =f(P◦P⁻¹(x_p)) = (f◦P)(x_R)^def= f_R(x_R),(f_R =f◦P), x_R =P⁻¹x_p. (1.8) From (1.7), we know that the inverse mapping P⁻¹ is continuous on Q_p\M_p

2 A Haar measure on Q and integration

In this section, a Haar measure is constructed by using the mapping P of R⁺∪ {0}intoQ_p\M_pand the Lebesque measure on R⁺∪ {0}. The symbol

Pis the set of all compact subsets of Q_p, andS is the σ−ring generated byP

.

Definition 2.1: LetE∈S, and put E_p=E\M_p, and E_R =P⁻¹(E_p). If E_R is a measurable set onR⁺∪ {0}, then we callEa measurable set onQ_p, and define a set functionµ_p(E) onS:

µ_p(E) = 1

pµ(E_R) (2.1)

whereµ(E_R)is the Lebesque measure on E_R. Thisµ_p(E) is called the mea- sure onE.

By the Definition 2.1, some examples can be given immediately:

(1) Leta_p, b_p∈Q_p, then µ_p{[a_p, b_p]}= (b_R −a_R)/p(see (1.7)) (2) µ_p{B_r(0)}=p^r (see (1.4))

(3) µ_p{S_r(0)}=p^r(1−¹_p) (see (1.5))

(4) Let{B_ri(a_i)}_ibe disjoint discs coveringE, by the definition of measure µ_pand definition of Lebesque exterior measure onR⁺∪{0}, it is evident that

µ_p(E) = inf

ri∈Zµ_p{∪_iB_ri(a_i)} (2.2) (5) µ_p(M_p) = 0.

It is obvious thatµ_p, by Definition 2.1, is countably additive. In order to prove thatµ_p is a Haar measure, we will give the following lemma.

Lemma 2.2: Ifα∈Q_p, then

µ_p{B_r(α)}=µ_p{B_r(0)}. (2.3)

Proof: 1^◦ Let α = p^−r1, for r₁ > r, r₁, r∈Z, and put x = p^−r1 + p^−rP

0≤xk<∞x_kp^k, x₀6=0,0≤x_k < p. ThenEis the set of all thesep-adic num- bers when x_k change fork = 0,1, ..., p−1. We write E_p ={x_p|x_p∈E\M_p}.

Forx_p∈E_p, let

P⁻¹(x_p) =p^r1+p^r X

0≤K<∞

x_kp^−k (2.4)

thenM_R = [p^r1, p^r1+p^r+1)is the set of all real numbers as presented in (2.4) (see(1.4)). Hence

µ_p(α+B_r(0)) =µ_p(B_r(α)) = 1

pµ(E_R) =p^r =µ_p(B_r(0)) (2.5)

2^◦Let α=p^−r2, forr₂≤r, r₂, r∈Z. thenα+B_r(0) = B_r(α) =B_r(0), by α∈B_r(0). So that

µ_p(B_r(α)) =µ_p(B_r(0)) (2.6) 3^◦Let α=p^−r3P

0≤k<∞α_kp^k, and put αⁿ=p^−r3P

0≤k≤nα_kp^k, applying to the result of1^◦ and 2^◦repeatedly in this case, we have

µ_p(αⁿ+B_r(0)) =µ_p(B_r(αⁿ)) =µ_p(B_r(0)) (2.7) However

n→∞lim µ_p(B_r(αⁿ)) = lim

n→∞

1

pµ{P⁻¹(B_rp(αⁿ))} (2.8) where B_rp(αⁿ) = B_r(αⁿ)\M_p. By the continuity of the mapping P⁻¹ (see(1.7)), we obtain

n→∞lim 1

pµ{P⁻¹(B_rp(αⁿ))}= 1

p{P⁻¹µ{P⁻¹(B_rp(α))}=µ_p{B_r(α)} (2.9) The part 3^◦follows from (2.7), (2.8) and (2.9).

Theorem 2.3: (The translation invariance of the measureµ_p) LetE∈Sand letα∈Q_p, then

µ_p(α+E) =µ_p(E) (2.10)

Proof: Let{B_ri(a_i)}^∞_i=1be disjoint discs coveringE, then{B_ri(a_i+α)}^∞_i=1 are disjoint discs coveringα+E. By the formula (2.2) in the example 4, we have

µ_p(α+E) = inf

ri∈Zµ_p{∪{α+B_ri(a_i)} (2.11) Applying the lemma 2.2 to the right side of the above formula, then

rinfi∈Zµ_p{∪_iB_ri(α+a_i)}

= inf

ri∈Z

X

i

µ_p{B_ri(α+a_i)}

= inf

ri∈Z

X

i

µ_p{B_ri(a_i)}

= inf

ri∈Zµ_p{∪_iB_ri(a_i)}

= µ_p(E)

Therefore,µ_p is a Haar measure.

According to the above definition of Haar measure, we can define the integration over measurable sets E in Q_p (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]))

Z

E

f(x_p)dµ_p (2.12)

By the theorem 2.3, the definition of measure and (1.8), we have the following theorem

Theorem 2.4: Supposef(x_p)is a complex-valued function onQ_p, thenf(x_p) is integrable over the interval [a_p, b_p] (a_p, b_p ∈ Q_p), 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

Z

[ap,bp]

f(x_p)dµ(x_p) = 1 p

Z 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) = x_p,P(a_R) = a_p,P(b_R) = b_p, a_p, b_p∈M¯ _p

Corollary 2.5: If f(x_p) is a bounded continuous function on the interval [a_p, b_p]⊂Q_p, then f(x_p) is integrable over[a_p, b_p], where [a_p, b_p] can be Q_p.

Notice that under the condition of theoremf_R(x_R)is a bounded piecewise continuous function on R⁺ ∪ {0} by (1.4), By the theorem 2.4, f(x_p) is integrable.

3 The indefinite integral and derivative of complex-valued function in Q

Definition 3.1: Let f be a complex-valued function defined inQ_p and for

∀x_p∈Q_p, f is integrable on interval [a_p, b_p], then f(x_p) =

Z xp 0

gdx_p (3.1)

is called on indefinite integral ofg.

Definition 3.2: Let f be a complex-valued function defined inQ_p, if there exist an integrable complex-valued functiongsuch that

f(x_p) = Z xp

0

gdx_p, x_p∈Q_p (3.2)

theng(x_p)is called the derivative off, which we will denote asf⁰(x_p).

In formula (3.2), letf = 1then µ([0, x_p]) =

Z xp 0

dµ (3.3)

The equation (3.3) follows that

¯

µ⁰(x)^def= µ⁰([0, x_p]) = 1 (3.4) Theorem 3.3: For complex-valued functions f, h on Q_p, if (f)_R(x_R) and (h)_R(x_R) are absolutely continuous, then

f⁰

R(x_R) = (f⁰)_R(x_R)

(f(x_p)h(x_p))⁰ =f⁰(x_p)h(x_p) +f(x_p)h⁰(x_p) (3.5)

Proof: Let f⁰ = g and g(x_p) = g(P(x_R)) = (g◦P)(x_R) = g_R(x_R). By definition (3.2) and theorem 2.4, we have

f(x_p) = Z xp

0

g(x_p)dx_p

= Z x_R

0

g_R(x_R)dx_R

= f_R(x_R) and therefore

(f_R)⁰(x_R) =g_R(x_R) =g(x_p) =f⁰(x_p) = (f⁰)_R(x_R) (3.6) and

(f(x_p)h(x_p))⁰ = (f_R(x_R)h_R(x_R))⁰

= (f_R)⁰(x_R)h_R(x_R) +f_R(x_R)(h_R)⁰(x_R)

= (f⁰)_R(x_R)h_R(x_R) +f_R(x_R)(h⁰)_R(x_R)

= f⁰(x_p)h(x_p) +f(x_p)h⁰(x_p)

From (3.6) it follows that

Corollary 3.4: If a complex-valued functionh(x_R) is absolutely continuous onR⁺∪ {0}, then f(x_p)^def= (hP⁻¹)(x_p)is derivable on Q_p\M_p.

Corollary 3.5: A locally constant function is derivable on Q_p\M_p, and its derivative is equal to 0.

Similarly, we can prove

Theorem 3.6: If f is derivable on [a_p, b_p],then Z bp

ap

f⁰(x_p)dµ=f(b_p)−f(a_p) (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 numbersQ_p.

Definition 4.1: Letf be a complex-valued function of p-adic variable. We define the centert_f of the graph{(x_p, f(x_p))|x_p∈Q_p} by

t^(R)

f def=

Z

Qp\Mp

P⁻¹(x_p)|f(x_p)|²dx_p/ Z

Qp\Mp

|f(x_p)|²dx_p

t_f = P(t^(R)

f )











(4.1)

if the integral (4.1) exists.

Definition 4.2: For a complex-valued function ofp-adic variable,we define the width off by

4_f = Z

Qp

|x_p−t_f|²|f(x_p)|²dx_p/ Z

Qp

|f(x_p)|²dx_p

!1/2

(4.2) if the integral (4.2) exists.

Theorem 4.3: LetP(t⁽_f^R)−a_R) =P(t⁽_f^R))−P(a_R),a_R =P⁻¹(a), a∈Q_p\M_p. (1) Iff is increasing, thent_Taf =t_f −a

(2) Supposesuppf⊂B_r(0). For a=p^−β, if β > r, then t_Taf =t_f −a (3) For a=p^−β, β∈Z, then

t_saf =at_f, where T_af(x_p) =f(x_p+a), S_af(x_p) =f(^x_a^p).

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(x_p), x_p∈Q_p\M_p. Hence

t^(R)

Taf = Z

Qp\Mp

P⁻¹(x_p)|T_af(x_p)|²dx_p/ Z

Qp

|T_af(x_p)|²dx_p

= Z

Qp\Mp

P⁻¹(x_p)|f(x+a)|²dx_p/ Z

Qp

|f(x_p)|²dx_p

≤ Z

R⁺

x_R|(f◦P)(x_R+a_R)|²dx_R/ Z

Qp

|f(x_p)|²dx_p

= Z

R⁺

(x_R −a_R)|(f◦P)(x_R)|²dx_R/ Z

Qp

|f(x_p)|²dx_p

= −a_R + Z

R⁺

x_R|(f◦P)(x_R)|²dx_R/ Z

Qp

|f(x_p)|²dx_p

= −a_R + Z

Qp\Mp

P⁻¹(x_p)|f(x_p)|²dx_p/ Z

Qp

|f(x_p)|²dx_p

= −a_R +t^(R)

f (4.3)

where we used µ(M_p) = 0, and for x_p, a∈B_r(0)∩(Q_p\M_p), x_p+a∈B_r(0).

On the other hand, using inequationP⁻¹(x−a)≥P⁻¹(x)−P⁻¹(a), we can easily obtain

t^(R)

Taf≥t⁽_f^R)−a_R (4.4)

From (4.3) and (4.4), we have t^(R)

Taf =t^(R)

f −a_R.

Finally, from the condition of (1) in theorem, we have

t_Taf =P(t⁽_f^R)−a_R) =P(t⁽_f^R))−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, t_f satisfy the condition of theorem 3.3, then

4_Taf =4_f (2) 4_Saf =|a|_p4_f

Proof: For (1), we have

4_Taf = Z

Qp

|x_p−t_Taf|²_p|T_af|²(x_p)dx_p/ Z

Qp

|T_af|²(x_p)dx_p

!1/2

= Z

Qp

|x_p−(t_f −a)|²_p|f(x_p+a)|²dx_p/ Z

Qp

|f(x_p+a)|²dx_p

!1/2

= Z

Qp

|t_p−t_f|²_p|f(t_p)|²dt_p/ Z

Qp

|f(t_p)|²dt_p

!1/2

=4_f

(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 inQ_pand the width of its Fourier transform. This theorem is similar to the Heisenberg uncertainty relation in quantum mechanics.

5 Main theorem

Lemma 5.1: Letx_p∈Q_p, then µ([0, x_p])≤ |x_p|_p

Proof: For x_p∈P_p\M_p x_p=p^−r

∞

X

k=0

x_kp^k∈Q_p, x₀6= 0,0≤x_k≤p−1

and therefore, we have P⁻¹(x_p) =p^r−1

∞

X

k=0

x_kp^−k≤p^r−1(p−1)

∞

X

k=0

p^−k=|x_p|_p (5.1)

By definition of measure µ_p, we have 1

pP⁻¹(x_p) =µ([0, x_p]) which leads to

µ(x_p)^def= µ([0, x_p])≤ |x_p|_p/p (5.2)

Theorem 5.2: Let f be complex-valued function of p-adic variable. If f∈L²(Q_p), f⁰∈L²(Q_p) and

lim

|bp|p→∞|b_p|_p|f(b_p)|²= 0, f(0) = 0 (5.3) then the following inequality is valid:

1

4π ≤ 4_f4

fb (5.4)

where fbis the transform off,

f(ξb _p) = Z

Qp

f(x_p)exp(2πi{ξ_px_p})dx_p

and by means of representation (1.1),{x_p} is defined as {x_p}=

0 if r(x_p)≥0 or x_p= 0 p^r(x₀+x₁p+· · ·+x_|r|−1p^|r|−1) if r(x_p)<0

Inequality (5.4) is called the Heisenberg uncertainty relation in harmonic analysis on p-adic numbers field.

Proof: By using (3.4) and theorem 3.3, we have

¯

µ(x_p−t_f)|f(x_p)|²0

=

¯

µ(x_p−t_f)f(x_p)χ_p(t

fbx_p)f(x_p)χ_p(t

fbx_p) 0

=|f(x_p)|²+ ¯µ(x_p−t_f) (f(x_p)χ_p(t

fbx_p)0

f(x_p)χ_p(t

fbx_p) +¯µ(x_p−t_f)f(x)χ_p(t

fbx_p)[f(x_p)χ_p(t

fbx_p)]⁰ (5.5) Therefore, from (3.7) we have

Z bp 0

|f(x_p)|²dx= ¯µ(x_p−t_f)|f(x_p)|²|^b₀^p

− Z bp

0

¯

µ(x_p−t_f)f(x_p)χ_p(t

fbx_p)

f(x_p)χ_p(t

fbx_p) 0

dx_p

− Z bp

0

¯

µ(x_p−t_f)f(x_p)χ_p(t

fbx_p)[f(x_p)χ_p(t

fbx_p)]⁰dx_p (5.6) where the functionχ_p(t

fbx_p) = exp(2πi{t

fbx_p})

By taking the limit of (5.5) as |b|_p→ ∞and using (5.2),(5.3), we obtain Z

Qp

|f(x_p)|²dx_p

≤ 2 Z

Qp

(¯µ(x_p−t_f))²|f(x_p)|²dx_p

!1/2

Z

Qp

|[f(x_p)χ_p(t

fbx_p)]⁰|²dx_p

!1/2

= Z

Qp

(¯µ(x_p−t_f))²|f(x_p)|²dx_p

!1/2

Z

Qp

|[f(·)χ_p(t

fb·)]^0∧(ξ)|²dξ

!1/2

= 2 Z

Qp

¯

µ(x_p−t_f)2

|f(x_p)|²dx_p

!1/2

Z

Qp

4π²|ξ|²|fb(ξ+t

fb)|²dξ

!1/2

(5.7) where we used the Hölder inequality for the integral andf⁰(·)^∧(ξ) =−2πiξfˆ(ξ), (f, f)_L²_(Qp)= (f ,bfb)_L²_(Qp), (f(·)χ_p(a·))^∧(ξ) =fb(ξ+a) From (5.2), we have

1 4π ≤

Z

Qp

|x_p−t_f|²_p|f(x_p)|²dx_p/ Z

Qp

|f(x_p)|²dx_p

!1/2

·

Z

Qp

|ξ_p−t

fb|²_p|f(ξb _p)|²dξ_p/ Z

Qp

|fb(ξ_p)|²dξ_p

!1/2

=4_f4_fb (5.8)

Hence we have completed our proof.

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Annales Mathematiques Blaise Pascal, 10:297–303, 2003.

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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