Weyl-Heisenberg frame in <span class="mathjax-formula">$p$</span>-adic analysis

ANNALES MATHÉMATIQUES

BLAISE PASCAL

Minggen Cui, Xueqin Lv

Weyl-Heisenberg frame in p-adic analysis

Volume 12, n^o1 (2005), p. 195-203.

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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 fieldQ_p, and introduce the definition and method of Weyl-Heisenberg frame on hormonic analysis top-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} → Q_p(f ield of p−adic numbers). This paper considers on the basis of the mappingP, gives the Weyl-Heisenberg frame in field ofp-adic numbers.

The field Q_p of thep-adic numbers is defined as the completion of field Qof rational with respect to the p-adic metric induced by thep-adic norm

|.|_p,(see [5]). Ap-adic numberx6= 0is uniquely represented in the canonical form

x=p^−r

∞

X

k=0

x_kp^k, |x|_p=p^r (1.1) where p is prime and r ∈Z (Z is integer set) , 0≤ x_k ≤p−1, x₀ 6= 0. For x, y ∈Q_p, 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, x_j < y_j from viepoint of (1.1).By interval[a, b], we mean the set defined by {x∈Q_p|a ≤ x≤b}.

It is known that if x=p^rPn

k=0x_kp^−k ∈R⁺∪ {0}and x₀6= 0, 0≤x_k ≤ p−1, k = 1,2,· · ·,then there is another expression;

x=p^r(

n−1

X

k=0

x_kp^−k+ (x_n−1)p⁻ⁿ+ (p−1)

∞

X

k=n+1

p^−k) (1.2)

M. G. Cui and L. Q. Lv

But we won’t use that expression (1.2) in this paper.

A mapping P : R⁺ ∪ {0} → Q_p was introduced in [4],[3],[2], as for x=p^rP∞

x=0x_kp^−k ∈R, x₀6= 0, 0≤x_k≤p−1, k= 1,2,· · · P(x) =p^−r

∞

X

k=0

x_kp^k

LetM_p=P(M_R), M_R ={x_R|x_R =p^r

n−1

X

k=0

x_kp^−k+ (n−1)p⁻ⁿ+ (p−1)

∞

X

k=0

p^−k, n∈Z⁺∪ {0}}

In the following to distinguish between real number field R and p-adic number fieldQ_p, number in Rdenotes by the subscriptR, and number with the subscriptpbelongs toQ_p. For example x_R, a_R, b_R inR; x_p, a_p inQ_p.

In [2] a measure is constructed using the mappingP fromR⁺∪ {0} into Q_p\M_p and Lebesgue measure onR⁺∪ {0}, the symbol P

is the set of all compact subsets ofQ_p, andS is theσ−ringgenerated byP

.

Definition 1.1:^[2] LetE ∈S, and putE_P =E\M_P, andE_R=P⁻¹(E_P).If E_Ris a measurable set onR⁺∪ {0}, the we callEis a measurable set onQ_p, and define a set functionµ_p(E)onS

µ_p(E) = 1 pµ(E_R)

where µ(E_R) is the Lebesgue measure on E_R.This µ_p(E) is called the measure onE.

By the Definition 1.1, some examples can given immediately:

(1)Let a_p, b_p∈Q_P, then µ_p{[a_p, b_p]}= (b_R−a_R)/p

(2)Let B_r(0) ={x_p||x_p|_p≤p^r, x_p∈Q_p}, then µ_p{B_r(0)}=p^r (3)Let S_r(0) ={x_p||x_p|_p=p^r, x_p∈Q_p}, then µ_p{S_r(0)}=p^r(1−_P¹) (4)µ_p{M_p}= 0

According to the above definition 1.1 of measure, we can define integra- tion over measurable setsE inQ_p

Z

E

f(x_p)dµ_p(x_p) or Z

E

f(x_p)dx_p

By the definition 1.1 of measure we have following theorem.

Theorem 1.2:(see [2]) Suppose f(x_p) is a Complex-Valued function onQ_p, thef(x_p)is integrable over the interval [a_p, b_p](a_p, b_p∈Q_p), if and if the real function f_R(x_R) defined onR⁺∪ {0} is Lebesgue integrable over the interval [a_R, b_R], and

Z

[ap,bp]

f(x_p)dx_p= 1 p

Z b_R

a_R

f(x_R)dx_R (1.3)

wheref_R(x_R) is defined

f(x_p) =f(P ◦P⁻¹(x_p)) = (f◦P)(x_R)^def= f_R(x_R), x_p=P(x_R)∈Q_p\M_p

2 Weyl-Heisenberg frame on p-adic number field

In real analysis, if there exists constants A and B,A, B >0, such that Akfk²_L2(R) ≤X

m,n

|(f, g_m,n)_L2(R)|²≤Bkfk²_L2(R)

holds for∀f ∈L²(R), theng_mn(x)is called the Weyl-Heisenberg frame, where g∈L²(R),p₀, q₀∈R,g_mn(x) =g(x−nq₀)e^2πimp0x, m, n∈Z and(f, g_mn)_L²_(R) is inner product inL²(R),

(f, g_mn)_L2(R)= Z

R

f(x)g_mn(x)dx.

In this section we give the definition of a Weyl-Heisenberg frame inQ_p by g_mn(x_p) =g(α_mn(x_p)−x_p) exp(2πimp₀ρ(x_p)) (2.1) where

α_mn(x_p) =P(|x_R+nq₀|) +x_p (2.2) and m, n, p₀, q₀ ∈ Z, x_R = P⁻¹(x_p),x_p ∈ Q_p\M_p . If g_mn(x_p) satisfies the frame condition :

Akfk²_L2 ≤X

m,n

|(f, g_m,n)_L2|²≤Bkfk²_L2, A, B >0,∀f ∈L²(Q_p)

M. G. Cui and L. Q. Lv

then we have f(x_p) =X

m,n

(f, g^∗_m,n)

L²g_m,n(x_p) =X

m,n

(f, g_m,n)g^∗_m,n(x_p), x_p∈Q_p where{g_m,n^∗ }is the dual frame of {g_m,n}:

g^∗_m,n=S⁻¹g_m,n andS is the frame operator:

Sf =X

m,n

(f, g_m,n)

L2g_m,n and

(f, g_mn)_L2 = Z

Qp

f(x_p)g_mn(x)dx

Theorem 2.1: Suppose f, g ∈L²(Q_p) are complex-valued functions defined on Q_p , p₀ = p^rp, q₀ = p^rq, r_p, r_q ∈ Z, if support cg_R(|w|) ⊂ [_2q⁻¹

0,_2q¹

0], and

∃A, B >0, such that A≤ X

m∈Z

|cg_R(|w−mp₀|)|²≤B, w∈R ,∀w 6= 0,

then functions g_mn(x_p) defined by (2.1) construct a frame of L²(Q_p), where g_R(|t|) = (g◦P)(|t|) =g(x_p),(P(|t|) =x_p, g_R =g◦P), t∈R, and cg_R(|w|) is the Fourier transform of g_R(|t|) in real analysis

cg_R(|w|) = Z

R

g_R(|t|) exp(−2πiwt)dt

Proof. From formula (2.2) org₍α_mn(x_p)−x_p) = (g◦P)(|x_R +nq₀|) = g_R(|x_R+nq₀|), for∀f ∈L²(Q_p), we have

X

m,n∈Z

|(f, g_m,n)_L2|²= X

m,n∈Z

| Z

Qp

f(x_p)g_m,n(x_p)dx_p|²

= X

m,n∈Z

| Z

Qp

f(x_p)g(α⁽ⁿ⁾_m (x_p)−x_p) exp(−2πimp₀ρ(x_p))|²

= 1 p

X

m,n∈Z

| Z

R⁺

f_R(x_R)g_R(|x_R+nq₀|) exp(−2πimp₀x_R)dx_R|² (2.3) where we used (1.3) and f_R(x_R) = (f◦P⁻¹)(x_p) in section 1

Let

f⁺

R(x) =

f_R(x), x≥0 0, x <0 From (2.3), we obtain

X

m,n∈Z

|(f, g_m,n)|²

= X

m,n∈Z

| Z

R

f_R⁺(x)(g_R(|x+nq₀|) exp(−2πimp₀x)dx|²

= X

m,n∈Z

| Z

R

fc_R⁺(w)[g_R(| ·+nq₀|) exp(2πimp₀·)]^∧(w)dw|² (2.4) where sign“·”is the argument on the function, for Fourier transform. But

[g_R(| ·+nq₀|) exp(2πimp₀·)]^∧(w)

=cg_R(|w−mp₀|) exp(2πinq₀(w−mp₀)) Hence from the support ˆg ⊂[−_2q¹

0,_2q¹

0] in condition of the theorem and (2.4) we have

X

m,n∈Z

|(f, g_mn)|²

= X

m,n∈Z

| Z

R

fc_R⁺(w+mp₀)cg_R(|w|) exp(−2πinq₀w)dw|²

= X

m,n∈Z

| Z _2q¹

0

−_2q¹ 0

fc_R⁺(w+mp₀)cg_R(|w|) exp(−2πinq₀w)dw|² (2.5)

M. G. Cui and L. Q. Lv

We know that c_n=q₀

Z _2q¹

0

−1 2q0

fc_R⁺(w+mp₀)cg_R(|w|) exp(−2πinq₀w)dw, n∈Z are Fourier coefficient of cf_R(w+mp₀)cg_R(|w|) on[_2q⁻¹

0,_2q¹

0

]. Hence by virtue of Parseval equality we have

P

n∈Z|c_n|²

=

∞

X

n=−∞

|q₀ Z _2q¹

0

−1 2q0

fc_R⁺(w−mp₀)cg_R(|w|) exp(−2πinq₀w)dw|²

=q₀ Z _2q¹

0

−_2q¹ 0

|fc_R⁺(w+mp₀)cg_R(|w|)|²dw

=q₀ Z

R

|fc_R⁺(w+mp₀)cg_R(|w|)|²dw

=q₀ Z

R

|fc_R⁺(w)cg_R(|w−mp₀|)|²dw (2.6) Comparing (2.5) and (2.6), we have

X

m,n∈Z

|(f_p, g_m,n)_L² |²= 1 q₀

Z

R

|fc_R⁺(w)|²G(w)dw (2.7) where

G(w) = X

m∈Z

|cg_R(|w−mp₀|)|²

Finally, by virtue of the conditions of the theorem, we have X

m∈Z,n∈Z

|(f, g_m,n)_L²|²=

( ≥_q^A

0

R

R|fc_R⁺(w)|²dw

≤ ^B

q₀

R

R|fc⁺

R(w)|²dw But

Z

R

|fc_R⁺(w)|²dw= Z

R

|f_R⁺(x)|²dx= Z

R⁺

|f_R(x_R)|²dx_R= Z

Qp

|f(x_p)|²dx_p=kfk_L². Hence we completed our proof.

3 Dual frame

In the section, we will give a formula to calculate the dual frame. By (2.7), we have

(Sf, f)_L² = X

m,n∈Z

|(f, g_m,n)_L²|²= 1 q₀

Z

R

fc_R⁺(w)G(w)cf_R⁺(w)dw

= 1 q₀

Z

R

(fc⁺

R(·)G(·))^∨(x)f_R⁺(x)dx

= 1 q₀

Z

R+

(fc_R⁺(·)G(·))^∨(x_R)f_R(x_R)dx_R where sign “ v " is the inverse Fourier transform. Therefore

(Sf, f)_L² = 1 q₀

Z

Qp

(fc_R⁺(·)G(·))^∨(P⁻¹(x_p))f(x_p)dx_p

= 1 q₀((cf⁺

R(·)G(·))^∨(P⁻¹(x_p)), f(x_p))_L2

Sincef is an arbitrary function inL²(Q_p), we conclude that (Sf)(x_p) = 1

q₀(fc_k⁺(·)G(·))^∨(P⁻¹(x_p)) or forx∈R⁺S{0}we conclude that

(Sf)_R(x_R) = 1 q₀(cf⁺

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⁺S{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 q₀fc⁺

R(w)G(w) (3.2)

M. G. Cui and L. Q. Lv

After replacef withS⁻¹f in formula (3.2), we have cf_R(w) = 1

q₀{(S⁻¹f)⁺_R}^∧(w)G(w) Which leads to

{(S⁻¹f)⁺_R}^∧(w) = q₀cf_R(w)

G(w) (3.3)

Then we take Fourier inverse transformation on both sides of (3.3), we have (S⁻¹f)⁺_R(x) ={q₀fc_R(·)

G(·) }^∨(x) So, forx≥0,

(S⁻¹f)_R(x_R) ={q₀f[_R(·) G(·) }^∨(x_R) is valied or

(S⁻¹)f(x_p) ={q₀f[_R(·)

G(·) }^∨(P⁻¹(x_p)) (3.4) Finally, letf(x_p) =g_mn(x_p)in formula (3.4) , we obtain

g_m,n^∗ (x_p) ={q₀f[_R(·)

G(·) }^∨(P⁻¹(x_p))

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.

[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