ISSN1842-6298 (electronic), 1843-7265 (print) Volume4(2009), 239 – 252
THE
n
-DIMENSIONAL CONTINUOUS WAVELET
TRANSFORMATION ON GELFAND AND SHILOV
TYPE SPACES
S. K. Upadhyay, R. N. Yadav and Lokenath Debnath
Abstract. In this paper the wavelet transformation on Gelfand and Shilov spaces of type WM(n),WΩ(∆n) andWMΩ(∆n) is studied. It is shown thatWψφ:WM(n)→WM(n×n+),
Wψφ:WΩ(∆n)→WΩ(∆n
×n+) andWψφ:WMΩ(∆n)→WMΩ(∆n×n+) is linear and continuous
wheren and ∆naren-dimensional real numbers and complex numbers. A boundedness result in
a generalized Sobolev space is derived.
1
Introduction
The spaces of typeWM(n),WΩ(∆n) andWMΩ (∆n) were investigated by Gelfand
and Shilov ([3]) and Friedman ([2]). It was shown that the Fourier transformation F : WM(n) → WΩ(∆n), F : WΩ(∆n) → WM(n) and F : WMΩ(∆n) → WΩ1
M1(∆
n) are linear and continuous. The main objective of this paper is to study
the continuous and wavelet transform of these W-type spaces. A complex-valued continuous functionψ with property
Z
n
ψ(t)dt= 0
is called awavelet.
Then-dimensional continuous wavelet transformation of functionφwith respect to waveletψ is defined by
e
φ(σ, a) = (Wψφ) (σ, a) = (2π)−n/2
Z
n φ(t)ψ
t−σ
a
dt
an/2 (1.1)
provided the integral exist fora= (a1, a2, a3,· · · , an)∈n+andσ= (σ1, σ2, σ3,· · · , σn)∈
n. If φ ∈ L2(n) and ψ ∈ L2(n) then by the Parseval formula of the Fourier
2000 Mathematics Subject Classification: 42C40; 46F12.
transformation, the wavelet transformation (1.1) can be defined in the Fourier space by the following formula:
e
φ(σ, a) = (Wψφ) (σ, a) = (2π)−n/2 Z
n
eihx,σiψb(ax)φb(x)dx. (1.2)
The reconstruction formula for (1.1) is given by
φ(x) = Wψ−1φ(σ, a)(x)
= Cψ−1(2π)−n/2
Z
n
+
Z
n e
φ(σ, a)ψ
x−σ a
da a2n
dσ
!
(1.3)
whereCψ =Rn
|ψ❜(x)|2
x dx >0.
We use [2] and [3] for the definitions of W-spaces of type WM(n), WΩ(∆n)
and WMΩ (∆n). Let Mj and Ωj be the convex functions such that
Mj(xj) = Z xj
0
µ(ξj)dξj (xj ≥0), (1.4)
Ωj(yj) = Z yj
0
ω(ηj)dηj (yj ≥0) (1.5)
forj= 1, 2, 3, · · ·,n. We set
µ(ξ) = (µ1(ξ1)),· · · ,(µn(ξn)), ω(η) = (ω1(η1)),· · ·,(ωn(ηn))
and
Mj(−xj) = Mj(xj), Mj(xj) +Mj x′j
≤Mj xj+x′j
(1.6) Ωj(−yj) = Ωj(yj), Ωj(yj) + Ωj y′j
≤Ωj yj +y′j
. (1.7)
The space WM(n) consists of allC∞-complex valued function φ(x) on n which
satisfy the inequalities
Dkxφ(x)
≤ Ckexp [−M(αx)], (1.8)
where exp [−M(αx)] = exp [−M1(α1x1)· · · −Mj(αjxj)· · · −Mn(αnxn)] and Ck, α >0 are constants which may depend on the function φ.
A function φ(z) ∈ WΩ(∆n) if and only if for β > 0 there exists a constant
Ck>0 such that zkφ(z)
and
exp [Ω (βy)] = exp [Ω [(β1y1) +· · ·+ Ω (βjyj) +· · ·+ Ω (βnyn)]],
where the constants Ck andβ depend on the function φ.
The space WMΩ (∆n) consists of all entire analytic functions φ(z) there exist constantsα >0,β >0 andC >0 such that
|φ(z)| ≤C[exp [−M(αx)] + Ω [(βy)]], z=x+iy (1.10)
where exp [−M(αx)] and exp [Ω [(βy)]] have usual meaning like (1.8) and (1.9) and constantsC,α and β depend on the functionφ.
Now we define the duality of functions M(x) and Ω (y) following the Young’s concept. Let M(x) and Ω (y) be defined by (1.6) and (1.7) respectively by the functionµ(ξ) andω(η) which occur in these equations are mutually inverse, that is
µ(ω(η)) = η and ω(µ(ξ)) = ξ, then the corresponding functions M(x) and Ω (y) are said to be dual in sense of Young. In this case, the Young inequality
xjyj ≤Mj(xj) + Ωj(yj) (1.11)
holds for any xj ≥ 0, yj ≥ 0, where equality holds if and only if yj = µj(xj) and xj varies in the interval x0j < xj < ∞, yj varies in yj0 < yj < ∞. The Fourier
duality relation is given byF[WM(n)] =WΩ(∆n),F
WΩ(∆n)=WM(n) and
FWMΩ (∆n)=WΩ1
M1(∆
n) from [2] and [3].
The present article is divided into three sections. Section 1 is an introduction with notations, auxiliary results of the wavelet transformation andW-type spaces. Section 2 gives the characterization ofW-spaces by using the concepts of the wavelet transformation. In the last section we derive some results of the wavelet transformation on Lp-Sobolev spaces.
2
Characterization of
W
-type spaces
In this section, we study the characterization of W-type space by using wavelet transformation
Lemma 1. Let ψb(z)∈WΩ(∆n) and φb∈WΩ(∆n). Then ψbφb∈WΩ(∆n).
This indicates that
zk+mψb(az)φb(z) = zkψb(az)zmφb(z)
= zkψb(az)zmφb(z)
≤ Ckexp [Ω [(αay)]]Cmexp [Ω [(ay)]]
≤ CkCmexp [Ω [α(a+ 1)y]]
≤ Ck,mexp [Ω [α(a+ 1)y]].
Theorem 2. Let the M(x) and Ω (y) be the functions, which are dual in sense of Young and ψ ∈ WM(n) and φ ∈ WM(n). Then the wavelet transformation Wψφ:WM(n)→WM n×n+
is continuous and linear.
Proof. The wavelet transformationWψφ with respect to waveletψ is given by
(Wψφ) (σ, a) = (2π)−n/2
Z
n
eihσ,xiψb(ax)φb(x)dx.
It follows from [2, p. 121.] that
(Wψφ) (σ, a) = (2π)−n/2
Z
n
eihσ,ziψb(az)φb(z)dx
= (2π)−n/2
Z
n
eihσ,ziψb(az)φb(z)dx.
For Φ (z, a) =ψb(az)φb(z). Therefore, (Wψφ) (σ, a) = (2π)−n/2 R
neihσ,ziΦ (z, a)dx,
wherez= (z1, z2, z3· · ·zn),a= (a1, a2, a3· · ·an) andz=x+iy.
Differentiating the above function with respect to σ and agives
DkσDma (Wψφ) (σ, a) = (2π)−n/2
Z
n
Then, applying Lemma1, we get
DkσDma (Wψφ) (σ, a)
≤ (2π)−n/2
Z
n e−σy
zkDamΦ (z, a)dx
(2.1)
≤ (2π)−n/2
Z
n
e−σy|z|
k+2+|z|k
(x2+ 1)n |D
m
a Φ (z, a)|dx !
≤ (2π)−n/2
exp (−σy)
Z
n
(Ck+2+Ck) exp [Ω [(a+ 1)αy]] dx
(x2+ 1)n
≤ (2π)−n/2exp [−σy+ Ω [(a+ 1)αy]] (Ck+2+Ck)Cn
≤ (2π)−n/2exp [−σy+ Ω [(a+ 1)αy]] (Ck+2,n+Ck,n)
≤ (2π)−n/2Dk,nexp [−σy+ Ω [(a+ 1)αy]]. (2.2)
Since y is found an arbitrary number then using [2, p. 22], we choose the sign ofy
in such a way that the following equality holds
|σ| |y|=M
σ
(a+ 1)α
+ Ω [(a+ 1)αy].
The exponent in the expression (2.2) becomes
−σy+ Ω [(a+ 1)αy] =−M
σ
(a+ 1)α
.
Then we get
DσkDam(Wψφ) (σ, a)
≤Ek,nexp
−M
σ
(a+ 1)α
.
This implies that
(Wψφ) (σ, a)∈WM n×n+
.
Lemma 3. Let ψ(ax)∈WΩ(∆n). Then Dβxψb(ax)∈WM(n).
Proof. We obtain from [3]
b
ψ(ax) = (2π)−n/2
Z
n
eihax,siψ(s)dσ,
Then
Dβxψb(ax) = (2π)−n/2
Z
n
(ias)βeihax,siψ(s)dσ
= (2π)−n/2
Z
n
(ias)βeihax,σ+iτiψ(s)dσ
= (2π)−n/2
Z
n
(ias)βeihax,σiψ(s)dσ.
Hence
Dβxψb(ax)
≤ (2π)−n/2
Z
n
|a|β|s|βe−axτ|ψ(s)|dσ
≤ (2π)−n/2
Z
n
1 +|a|β sβψ(s)
exp (−axτ)dσ
≤ (2π)−n/2
Z
n
1 +|a|β 1 +s2sβψ(s)
exp (−axτ) dσ (1 +σ2)n/2.
From (1.7) and (1.9) we obtain
Dβxψb(ax)≤Cn
1 +|a|β(Cβ +Cβ+2) exp [Ω (βτ)−axτ].
Using (1.11) gives
Dβxψb(ax)≤(2π)−n/2Cn
1 +|a|β(Cβ+Cβ+2) exp
−M
ax β
.
Then we obtain
exp
M
ax
β
Dβxψb(ax)
≤Cn,α,β
1 +|a|β.
Lemma 4. Let φb(x)∈WM(n) andψb(ax)∈WM(n). Then
|DmxΦ (x, a)| ≤(1 +|a|m)Dmexp
−M
a−1
α
x
, m >0 and α >0,
Proof. We have
From Lemma3, the above expression yields
|DmxΦ (x, a)|
Proof. The wavelet transformation of a functionφwith respect to waveletψis given by
(Wψφ) (σ, a) = (2π)−n/2
Z
n
From [2, p. 20-21], we find that
Using Young inequality (1.11), the above integral can be evaluated as follows
|x| |τ| −M
Therefore, we have
|(is)m(Wψφ) (s, a)|
This implies that
Wψφ∈WΩ ∆n×n+
Lemma 6. Let φb∈WΩ1
M1(∆
n) andψb(az)∈WΩ1
M1(∆
n). Then
|Φ (z, a)| ≤Dexp [−M1{(a−1)αx}+ Ω1[(a+ 1)βy]].
Proof. Now,
|Φ (z, a)| = bφ(z)ψb(az)
=
bφ(z)
bψ(az)
≤ Cexp [−M1(αx) + Ω1(βy)]C′exp [−M1(aαx) + Ω1(aβy)].
From (1.6) and (1.7) we have
|Φ (z, a)| ≤ CC′exp [{−M1(aαx) +M1(αx)}+ Ω1[(βy) + Ω1(aβy)]]
≤ Dexp [−M1{(a−1)αx}+ Ω1{(a+ 1)βy}].
Therefore,
|Φ (z, a)| ≤Dexp [−M1{(a−1)αx}+ Ω1{(a+ 1)βy}] .
Theorem 7. Let Ω1(x) and M1(x) be the functions which are dual in the sense of
Young to the functions M(x) and Ω (x) and φ∈ WMΩ (∆n), ψ ∈ WΩ
M(∆n). Then the wavelet transformationWψφis a continuous linear mapping fromWMΩ (∆n)into WMΩ ∆n×n+
.
Proof. From (1.2) the wavelet transformation is given by
(Wψφ) (σ, a) = (2π)−n/2
Z
n
eihx,σiφb(x)ψb(ax)dx
= (2π)−n/2
Z
n
eihx,σiΦ (x, a)dx,
where Φ (x, a) =φb(x)ψb(ax). It follows from [2, p. 23] that
(Wψφ) (s, a)
= (2π)−n/2
Z
n
eihs,ziΦ (z, a)dz,
= (2π)−n/2
Z
n
exp [i(σ+iτ) (x+iy)] Φ (z, a)dx
= (2π)−n/2
Z
n
exp [i(−σy−τ x) +i(σx−τ y)] Φ (z, a)dx
So that
|(Wψφ) (s, a)|
≤ (2π)−n/2
Z
n
exp [(−τ x−σy)]|Φ (z, a)|dx
≤ (2π)−n/2D Z
n
exp [(τ x−σy)] exp [−M1(a−1)αx] + Ω1[(a+ 1)βy]dx
= (2π)−n/2Dexp [−σy+ Ω1{(a+ 1)βy}]
Z
n
exp [τ x−M1{(a−1)αx}]dx.
Now using the arguments similar to those of Theorem 2.2 and Theorem 2.5, we find that
|(Wψφ) (s, a)| ≤ Dexp
−M
σ β(1 +a)
+ Ω
τ
(a+ 2)
Z
n
exp [−M(αx)]dx
≤ D′exp
−M
σ β(1 +a)
+ Ω
τ
(a+ 2)
.
Hence
(Wψφ) (s, a)∈WMΩ ∆n×n+
.
3
Wavelet transformation on
L
p(
n
)
-Sobolev spaces
In this section we study the wavelet transformation of a functionφ∈WMΩ (∆n) with respect to waveletψ∈WMΩ (∆n) on Lp-type Sobolev space.
Definition 8. Form∈and1≤p <∞, the spaceGm,p(∆n) is defined to the set of allφ∈ WMΩ (∆n)′ such that
kφkGm,p = Z
n
1 +|s|2m/2bφ(s)pdσ 1
p ,
where φb=Fφ ands=σ+iτ.
Definition 9. We define the space of all measurable function φ onn+×∆n such that
eΦ (s, a)
Hpp′,m
=
"Z
n
+
Z
n
|Φ (s, a)|dσ p′/p
a−m−1da #
,
Lemma 10. Let φ∈WΩ
Proof. From Theorem5 the wavelet transformation of a functionψ∈WMΩ (∆n) can be written as
Using Lemma10 yields
"Z
n
+
am−1 Z
n
|(Wψφ) (s, a)|p′dσ p/p′
da #
≤ Dp′Cψm,p
Z
n
1 +|s|2m/2
bφ(s)
pdσ
= Dp′Cψm,pkφkGp,m.
Then
eΦ (s, a)
Hp,m p′
≤Eψm,p,p′kφkGp,m.
References
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0816.46033.
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S. K. Upadhyay R. N. Yadav
Department of Applied Mathematics, Department of Mathematics and Statistics, I. T. and C I M S, D S T, B. H. U., D. D. U. Gorakhpur University,
Varanasi - 221005, Gorakhpur,
India. India.
email: sk [email protected]
Lokenath Debnath
Department of Mathematics,
The University of Texas-Pan American, 1201 West University Drive,
