The Affine Frame in <span class="mathjax-formula">$p$</span>-adic Analysis

ANNALES MATHÉMATIQUES

BLAISE PASCAL

Ming Gen Cui, Huan Min Yao, Huan Ping Liu The Affine Frame in p-adic Analysis

Volume 10, n^o2 (2003), p. 297-303.

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The Affine Frame in p-adic Analysis

Ming Gen Cui Huan Min Yao Huan Ping Liu

Abstract

In this paper, we will introduce the concept of affine frame in wavelet analysis to the field of p-adic number, hence provide new mathematic tools for application of p-adic analysis.

1 Introduction

The concept of affine frame was introduced to wavelet analysis at first in reference [2]. And the theory of affine frame was studied in reference [3] and [4] further. In this paper, we introduce the concept of affine frame in wavelet analysis to the field of p-adic number as the discrete formula of wavelet transform. Before now, we have introduced the wavelet transform to the field of p-adic analysis([1]). General knowledge on p-adic analysis see [6].

It is known that if x = p^rPn

k=0x_kp^−k ∈ R⁺∪ {0}, x₀ 6= 0. 0 ≤ x_k ≤ p−1, k = 1,2,· · · ,then there is another expression for x.

x=p^r(

n−1

X

k=0

x_kp^−k+ (x_n−1)p⁻ⁿ+ (p−1)

∞

X

k=n+1

p^−k) (1.1) We don’t adopt this expression. LetM_R be the set of numbers in the form of (1.1). An mapping ρ was introduced in reference [5], which ρ : Q_p → R⁺∪ {0}, forx=p^−rP∞

x=0x_kp^k, x₀6= 0, 0≤p−1, k= 1,2,· · · ρ(x) =p^−r−l

∞

X

k=0

x_kp^−k (1.2)

Fora_R, b_R ∈R⁺∪ {0}, mapping (1.2) follows that

b_R −a_R =µ([a_p, b_p]), a_p=ρ⁻¹(a_R), b_p=ρ⁻¹(b_R)

M.G. Cui, H.M. Yao and H.P. Liu

and then we have the following lemma.

Lemma 1.1:Letf(x_p)∈L²(Q_p)andf_R(x_R)^def= f(ρ(x_p)), x_p∈Q_p, , x_R = ρ(x_p)then

Z bp ap

f(x_p)dx_p= Z b_R

a_R

f_R(x_R)dx_R, where a_R =ρ(a_p), b_R =ρ(b_p).

Remark: µ

a_p, b_p]} is defined as the infimum of measure of all the disjoint discs covering

B_ri(a_i) for interval[a_p, b_p],and for complex-valued functionf, the integral of f is defined by

Z

[ap,bp]

fdx_p^def= inf_{

Bri(ai)}

X

i

f(a_i)µ B_ri(a_i) .

2 The Frame in L

(Q

)

Letf, h∈L²(Q_p).The following we will discuss conditions of{h_m,n}becomes to the frame ofL²(Q_p).Here, the so-called frame defined as: ∃A, B >0such that

Akfk²_L2(Qp) ≤X

m,n

|(f, h_mn)|²_L2(Qp) ≤Bkfk²_L2(Qp)

where

kfk²_L2(Qp) = Z

Qp

|f|²dx_p

(f, h_mn)_L²_(Qp) = Z

Qp

f h_mndx_p

It is known that if {h_mn} is the frame of L²(Q_p), then the function f ∈ L²(Q_p)can be express as

f(x_p) =X

m,n

(f, h^∗_mn)_L2(Qp)h_mn(x_p) =X

m,n

(f, h_mn)_L2(Qp)h^∗_mn (2.1)

here h^∗_mn = S⁻¹h_mnis the dual frame of h_mn,and S : L²(Q_p) → L²(Q_p) is the frame operator.

Sf =X

m,n

(f, h_mn)_L2(Qp)h_mn

Theorem2.1. Letf, h∈L²(Q_p)be complex-value function, andh(0) = 0, x_R =ρ(x_p), x_p∈Q_p\M, hereM =ρ⁻¹(M_R), and measure of setM be equal to 0. Selectb₀=p^r(b0), r(b₀)∈Z, a₀=p^r(a0), r(a₀)∈Z. If

(1) ∃A, B >0such that forω6= 0 A≤G(ω)^def= X

m∈Z

bh₁( ω

a^m₀ )

2≤B,

(2) suppbh₁⊂[−_2b¹

0,_2b¹

0]

then (

h_mn(x_p)^def= a^m/2₀ h α^(m)n (x_p)−x_p a^−m₀

!)

n,m∈Z

is a frame ofL²(Q_p), where α^(m)_n (x_p) =

ρ⁻¹(x_R+a^−m₀ nb₀) +x_p , x_R+a^−m₀ nb₀>0 x_p , x_R+a^−m₀ nb₀≤0 the signb denotes the Fourier transform

fb(w) = Z

R

f(x)e^−2πiωxdx andh₁is defined from(2.3).

Proof. From the definition ofα_n^(m)(x_p), we have (Sf, f)_L2(

Qp) = X

m,n∈Z

(f, h_mn)_L²_(Qp)

2

= X

m,n∈Z

a^m₀ Z

Qp\M

f(x_p)h

(αn^(m)(x_p)−x_p a^−m₀

) dx_p

2

= X

m,n∈Z

a^m₀ Z

R⁺∪{0}

f_R(x_R)h₁

x_R+a^−m₀ nb₀ a^−m₀

dx_R

2

(2.2)

where x_R =ρ(x_p), x_p∈Q_p\M, f_R(x_R) =f(ρ⁻¹(x_R)), h₁(x_R) =

h(ρ⁻¹x_R), x_R >0

h(0), x_R ≤0 (2.3)

M.G. Cui, H.M. Yao and H.P. Liu

Let

f_R⁺(x_R) =

f_R(x_R), x_R ≥0 0, x_R <0 Thus, from(2.2), the following equalities hold

X

m,n∈Z

(f, h_mn)_L²_(Qp)

2

= X

m,n∈Z

a^m₀ Z

R

f_R⁺(x_R)h₁ x_R+a^−m₀ nb₀ a^−m₀

dx_R

2

= X

m,n∈Z

a^−m₀ Z

R

fc_R⁺(ω)hb₁( ω

a^m₀ )exp(2πia^−m₀ nb₀ω)dω

2

= X

m,n∈Z

a^−m₀

Z ^am_2b⁰

0

−^am_2b⁰ 0

fb_R⁺(ω)bh₁( ω

a^m₀ )exp(2πia^−m₀ nb₀ω)dω

2

(2.4)

In the last equality, we usedsupphb₁⊂[−^a_2b^m0

0,^a_2b^m0

0]. But

1 a^m₀/b₀

Z a^m₀ 2b₀

−a^m₀ 2b₀

fc_R⁺(ω)hb₁( ω

a^m₀ ) exp 2πia^−m₀ nb₀ω dω

is the Fourier coefficient of the function fc_R⁺(ω)hb₁( ω

a^m₀ ). Denotes this coeffi- cient byc−n. By the P arseval equality, we have

X

n∈Z

c_n

2= 1

a^2m₀ /b²₀ X

n∈Z

Z a^m₀ 2b₀

−a^m₀ 2b₀

fc_R⁺(ω)hb₁( ω

a^m₀ ) exp 2πia^−m₀ nb₀ω dω

2

= b₀ a^m₀

Z a^m₀ 2b₀

−a^m₀ 2b₀

fc_R⁺(ω)hb₁( ω a^m₀ )

2

dω (2.5)

By (2.4), we have

X

m,n∈Z

(f, h_mn)_L2(

Qp)

2

= 1 b₀

X

m∈Z

Z a^m₀ 2b₀

−a^m₀ 2b₀

fc_R⁺(ω)hb₁( ω a^m₀ )

2

dω

= 1 b₀

Z

R

|fc_R⁺(ω)|²|hb₁( ω

a^m₀ )|²dω (2.6)

Here

G(ω) = X

m∈Z

hb₁( ω a^m₀ )

2

(2.7)

Finally, by the condition of theorem 0< A≤ |G(ω)|² ≤B and formula (2.6), we have

X

m,n∈Z

|(f, h_mn)_L2(

Qp)|²=







≥ 1 b₀AR

R|fb_R⁺(ω)|²dω

≤ 1 b₀BR

R|fb⁺

R(ω)|²dω But

Z

R

|fb_R⁺(ω)|²dω= Z

R

|f_R⁺(x_R)|²dx_R = Z

R⁺∪{0}

|f_R(x_R)|²dx_R = Z

Qp

|f(x_p)|²dx_p

=kfk²_L2(Qp)

Hence, the theorem follows.

3 Dual frame h

It is known that if {h_mn}_mn construct a frame of L²(Q_p), then expression (2.1) is valid. From (2.6), we have

(Sf, f)_L²_(Qp) = 1 b₀

Z

R

|fb_R⁺(ω)|²G(ω)dω

= 1

b₀ Z

R

fc_R⁺G∨

(x_R)f⁺_R(x_R)dx_R

= 1

b₀ Z

R⁺∪{0}

(fb_R⁺G)^∨(x_R)f_R(x_R)dx_R

= 1

b₀ Z

Qp

fc⁺

RG∨

ρ(x_p)

f(x_p)dx_p (3.1)

where x_p = ρ⁻¹(x_R), x_R ∈ R⁺∪ {0}, and “∨" is the sign of Fourier inverse transform. Here, we use lemma A.

M.G. Cui, H.M. Yao and H.P. Liu

(3.1) can be rewritten as (Sf, f)_L2(Qp) = 1

b₀

( ˆf_R⁺G)^∨ ρ(x_p)

, f(x_p)

L²(Qp)

(3.2)

Sincef is an arbitrary function inQ_p, Sf

(x_p) = 1 b₀

fˆ_R⁺G∨

ρ(x_p)

(3.3)

or forx∈R⁺∪ {0}, we conclude that (Sf)_R(x_R) = 1

b₀

fˆ_R⁺G∨

(x_R), (Sf)_R =S(f◦ρ⁻¹) (3.4) On the basis of formula (3.4), we can extend the domain of function(Sf)_R(x_R) onto R. Therefore

(Sf)^∧_R(ω) = 1

b₀fc_R⁺(ω)G(ω), ω∈R⁺∪ {0} (3.5) Replacingf byS⁻¹f in the formula (3.5), we have

fc_R(ω_R) = 1

b₀(S\⁻¹f)⁺_R(ω)G(ω) Thus

(S\⁻¹f)⁺

R(ω_R) = b₀cf_R(ω) G(ω) or

(S⁻¹f)⁺_R(x_R) =b₀n cf_R G

o∨

(x_R)

Forx_R ≥0, we have

(S⁻¹f)_R(x_R) =b₀n cf_R G

o∨

(x_R) or

(S⁻¹f)(x_p) =b₀ cf_R G

∨

ρ(x_p)

So for f_R(x_R), x_R ≥0,(S⁻¹f)(x_p) =b₀[f_R∗(G⁻¹)^∨] is valid. Finally, We have

h^∗_mn(x_p) =b₀[(h_mn)_R∗(G⁻¹)^∨](ρ(x_p)), where the sign∗ denotes convolution.

References

[1] M.G. Cui. Note on the wavelet transform in the fieldq_pof p-adic numbers.

Appl. and Computational hormonic Analgsis, 13:162–168, 2002.

[2] I. Daubechies, A. Grossman, and Y. Meyer. Painless nonorthogonal ex- pansion. J. Math. Phys., 27:1271–1283, 1986.

[3] E. Ch. Heil and F. Walnut. Continuous and discrete wavelet transforms.

SIAM Review, 31:628–666, 1989.

[4] B. Lian, K. Liu, and S.-T. Yau. Mirror principle I. Asian J. Math., 4:729–763, 1997.

[5] S.V.Kozyrev. Wavelet theory asp-adic spectral analysis.Izv. Russ. Akad.

Nauk, Ser. Math., 66:149–158, 2002.

[6] V.S.Vladimirov, I.V.Volovich, and E.I.Zelenov.p-adic analysis and Math- ematical Physics. World Scientific, 38 -112, 1994.

Ming Gen Cui Huan Min Yao

Harbin Institute of Technology Harbin Normal University Department of Mathematics Department of Information

Wen Hua Xi Road Science

Weihai, Shandong He Xing Road

P.R. CHINA Harbin, Heilongjiang

[email protected] P.R. CHINA

[email protected]

Huan Ping Liu

Harbin Normal University Department of Information

Science He Xing Road

Harbin, Heilongjiang P.R. CHINA

[email protected]