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The isometric identities and inversion formulas of complex continuous wavelet transforms
Applied Mathematics and Computation 233 (2014) 116–126
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The isometric identities and inversion formulas of complex continuous wavelet transforms Shasha Li a,b,⇑, Caixia Deng c, Wei Sun a,c a
Institute of Mathematics, Jilin University, Changchun 130012, China Department of Mathematics, Daqing Normal University, Daqing 163712, China c School of Applied Sciences, Harbin University of Science and Technology, Harbin 150080, China b
a r t i c l e Keywords: Reproducing kernel Wavelet transform Image space
i n f o
a b s t r a c t The reproducing kernel function of the image space of a family of complex wavelet transforms is presented. An admissible wavelet is obtained by convolution computation. Next, the correlative characterisation of the image space of the family of complex wavelet transforms is provided when the scale is fixed. Furthermore, the isometric identities and inversion formulas are obtained, which provide a theoretic basis for investigating the image space of the general wavelet transform. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Wavelet analysis has been rapidly developed in the mathematical field. This analysis is a brilliant combination of pure mathematics and applied mathematics following the Fourier transform, which has been called a mathematics microscope. In addition, wavelet analysis is also one of the most obvious achievements of harmonic analysis in the past half century. Wavelet analysis can effectively provide useful information from signals, and it also can solve many difficult problems that cannot be solved using the Fourier transform. Currently, this analysis has made most of the important valuable achievements in, for instance, the field of signal analysis (see Ref. [1]), image processing (see Ref. [2]), sound processing (see Ref. [3]), and solutions of equations (see Refs. [4–7]). We all know that there is redundancy of information with wavelet transforms, and the wavelet transform coefficient has relevance in the wavelet transform image plane. We can also see that the reproducing the kernel Hilbert space is the basis of the wavelet transform, and the relevance region’s magnitude is given by the reproducing the kernel. In addition, we can prove that its magnitude will decrease if the scale decreases. Therefore, we can conclude that the reproducing the kernel space plays an important role in the reconstruction of the continuous wavelet transform (see Refs. [8–11]). With the development of wavelet analysis, reproducing kernel theory has attracted increasing attention from many scholars. For example, Saitoh described reproducing kernels of the direct product of two Hilbert spaces (see Refs. [12–14]). Deng and Du have described the image space of wavelet transform such as that of Shannon, Meyer and Littlewood-Paley (see Refs. [15–17]). Li et al. further studied the image space of the complex Gauss wavelet transform (see Ref. [18]). Castro et al. have identified a general discretisation method for solving wide classes of mathematical problems by applying the theory of reproducing kernels (see Refs. [19,20]). It can be observed that continuous wavelet transform is the foundation of the reproducing kernel function of the image space. Therefore, we can choose the most suitable wavelet bases according to the structure of the reproducing kernel, which is based on the information redundancy of the continuous wavelet transform. In this paper, we present a family of admissible complex wavelets and describe the spaces with the help of ⇑ Corresponding author at: Institute of Mathematics, Jilin University, Changchun 130012, China. E-mail addresses: [email protected] (S. Li), [email protected] (C. Deng), [email protected] (W. Sun). 0096-3003/$ - see front matter Ó 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.079
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
117
Fock kernel. The paper demonstrates isometry and inversion formulas under the fixed scale-factor conditions and further provides a theoretical foundation for the studies of the general wavelet transform. It is obvious that the results of Ref. [18] are a special case for this article. 2. Definition and properties In this section, we introduce the concept and properties involved with reproducing kernels and wavelets. Definition 2.1. Let H be a Hilbert function space. Its elements are real-valued or complex-valued functions on an abstract set E. The inner can be denoted by
hf ; gi ¼ hf ðÞ; gðÞi;
f; g 2 H
for any fixed q 2 E; Kðp; qÞ 2 H as a function in p, and for any f 2 H, and q 2 E, we have
f ðqÞ ¼ hf ðpÞ; Kðp; qÞi; then Kðp; qÞ will be called the reproducing kernel of the Hilbert function space H; H will be called the reproducing kernel Hilbert space (RKHS). Furthermore, the function Kðp; qÞ is uniquely determined by RKHS H. Therefore, it can be written as HK , that is, H ¼ HK . Furthermore, we can also write the Definition below for the reproducing kernel via linear transformation. Proposition 2.1. Let F ðEÞ be a linear space comprising all complex-valued functions on an abstract set E. Let H be a Hilbert space equipped with product h:; :iH . Let h; E ! H be a Hilbert space H-valued function on E. We consider the linear mapping which is L from Hinto F ðEÞas defined by
f ðpÞ ¼ ðLfÞðpÞ ¼ ðf; hðpÞÞH :
ð2:1Þ
We suppose that
kðp; qÞ ¼ hhðqÞ; hðpÞiH :
ð2:2Þ
Let RðLÞ be the range of L for H and we introduce the inner product in RðLÞ induced form the norm
kf kRðLÞ ¼ inf kfkH ;
f ¼ Lf:
Definition 2.2. For the Hilbert space F ðkÞ ðk > 0Þ comprising all entire functions f ðzÞ with finite norms
ZZ k
12 2 jf ðzÞj2 ekjzj dxdy < 1;
p
C
the function ekuz is the reproducing kernel. This space will be called the Fock space (or the Bargmann–Fock space; see Ref. [14]). Lemma 2.1. For a RKHS HK on E and for any non-vanishing complex-valued function sðpÞ on E,
ks ðp; qÞ ¼ sðpÞsðqÞkðp; qÞ;
p; q 2 E
is a reproducing kernel for the Hilbert space Hks , comprising all the functions fs ðpÞ on E that are expressible in the form
fs ðpÞ ¼ f ðpÞsðpÞ;
f 2 HK
and that is equipped with the inner product
hfs ; g s iHk ¼ s
fs g s ; ; s s HK
(see Ref. [12]). ^ xÞ satisfies the admissible condition Definition 2.3. Let wðtÞ 2 L2 ðRÞ , and its Fourier transform wð
Cw ¼
Z R
^ xÞj2 jwð dx < 1; j xj
then wðtÞ is called the basic wavelet or mother wavelet, and it is also known as an admissible wavelet.
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S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Definition 2.4. Let f ðtÞ 2 L2 ðRÞ; wðtÞ be a basic wavelet, and
1 tb ; wa; b ðtÞ ¼ pffiffiffiffiffiffi w a jaj where a; b 2 R; a – 0, then the function wa; b ðtÞ is a continuous wavelet that is generated by wðtÞ and is related to with parameters a and b, and
1 ðT wav f Þða; bÞ ¼ pffiffiffiffiffiffi jaj
tb dt f ðtÞw a R
Z
is the continuous wavelet transform of f ðtÞ. The continuous wavelet transform of f ðtÞ in a frequency domain is given by
ðT wav f Þða; bÞ ¼ hf ; wa; b i ¼
pffiffiffi Z a 1 ^ ^ a; b ^f ðxÞwða ^ xÞeibx dx: hf ; w i ¼ 2p R 2p
The continuous wavelet transform of f ðtÞ can also be written as a convolution type in the frequency domain as given by 1
ðT wav f Þða; bÞ ¼ jaj2 f wjaj ðbÞ
0
where
b : wjaj ðbÞ ¼ jaj1 w a Thus, a wavelet transform can be observed as a convolution computation of the signal and filter. wjaj ðbÞ can be comprehended as a high-pass filter from an engineering point of view. Lemma 2.2. For all f ðtÞ; gðtÞ 2 L2 ðRÞ, we have
hf ; gi ¼
1 Cw
Z Z R
ðT wav f Þða; bÞðT wav gÞða; bÞ R
1 dadb a2
where a; b 2 R; a – 0 (see Ref. [21]). Indeed, we have directly
Z Z R
ðT
wav
R
f Þða; bÞðT
wav
#" 1 Z # Z Z " 12 Z 2 jaj dadb 0 ib l 0 0 ^f ðlÞeibl wða 0 ^ lÞdl jaj ^ g^ðl Þe wðal Þdl a2 2 p 2 p R R R R Z Z Z da ^f ðlÞg^ðlÞjwða ^ lÞj2 dl ¼ C w f ðxÞgðxÞdx: ¼ 2p R jaj R R
dadb gÞða; bÞ 2 ¼ a
Thus, we find that the reproducing kernel of the image space of wavelet transform is given by
Kða0 ; b0 ; a; bÞ ¼ hwa;b ðxÞ; wa0 ; b0 ðxÞi ¼
Z
wa; b ðxÞwa0 ; b0 ðxÞdx:
R
The norm in the RKHS Hk is given by
1 Cw
Z Z R
jðT wav f Þða; bÞj2 R
12 1 dadb ; a2
2
where f 2 L ðRÞ. The finiteness of the above integral does not, however, characterise the members in Hk . That is, Hk forms a subspace in the Hilbert space comprising all functions on R R with above finite norms. Thus, we are interested in the characterisation of the subspace Hk . 3. The image space of a family of complex wavelet transform Theorem 3.1. Let wðtÞ be a basic wavelet, gðtÞ 2 L1 ðRÞ be a bounded function, and then its convolution w gðtÞ is also a basic wavelet. ^ðxÞ is bounded. By the Cauchy–Schwarz inequality, we have Proof. As gðtÞ 2 L1 ðRÞ; g
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Z Z w gðtÞj2 dt ¼
Z 2 2 Z Z wðt uÞgðuÞdu dt 6 jwðt uÞjjgðuÞjdu dt R R R R Z Z Z 6 jwðt uÞj2 jgðuÞjdu jgðuÞjdu dt
R
¼
R
R
R
R
Z Z
R
Z 2 Z Z jwðt uÞj2 dtjgðuÞj du jgðuÞjdu ¼ jwðtÞj2 dt jgðuÞjdu R
R
R
< þ1: Then, we can obtain w gðtÞ 2 L2 ðRÞ and
Z R
jw ^ gðxÞj2 dx ¼ j xj
Z R
^ xÞj2 jg^ðxÞj2 jwð dx 6 supjg^ðxÞj2 jxj
Z R
^ xÞj2 jwð dx < þ1: j xj
By Definition 2.4, thus, the function w xðtÞ is an admissible wavelet function. 2
h 2
Proposition 3.1. Let f ðtÞ ¼ eit2t , and gðtÞ ¼ f ðk1Þ ðtÞ, where k 2 Zþ is a positive integer, and w1 ðtÞ ¼ eit2t ði 4tÞ, then the convolution function wðtÞ ¼ w1 gðtÞ is a basic wavelet. Proof. The Fourier transform of w1 ðtÞ is
^ 1 ð xÞ ¼ w
pffiffiffiffiffiffiffi ð1þxÞ2 2p ixe 8 : 2
The Fourier transform of f ðtÞ is
pffiffiffiffiffiffiffi 2p ð1þxÞ2 e 8 : 2
^f ðxÞ ¼
The Fourier transform of gðtÞ is
g^ðxÞ ¼
pffiffiffiffiffiffiffi ð1þxÞ2 2p k1 ðixÞ e 8 : 2
Then, we can find that w1 ðtÞ satisfies the admissible condition, gðtÞ 2 L1 ðRÞ, and gðtÞ is bounded. So, we can see by Theorem 3.1 that
w ¼ w1 gðtÞ is a basic wavelet function.
h
The Fourier transform of wðtÞ is
^ xÞ ¼ w ^ 1 ðxÞg^ðxÞ ¼ p ðixÞk eð1þ4xÞ : wð 2 2
Then, we can find the conclusion by the inversion formula of Fourier transform. When k ¼ 1; wðtÞ below is the cgau wavelet (complex Gauss wavelet) function
wðtÞ ¼
pffiffiffi 2 2eitt ði 2tÞ:
When k ¼ 2; wðtÞ below is the complex Mexihat wavelet function
pffiffiffiffi 2 wðtÞ ¼ 2 peitt ð4t 2 þ 4it 3Þ: By the symmetry in a, we merely consider the wavelet transform for a > 0. We shall consider the wavelet transform in the complex form below
ðT wav f ÞðA; zÞ ¼
1 tz dt; f ðtÞA2 w A R
Z
ð3:1Þ
or 1 1 ðT wav f ÞðA; zÞ ¼ pffiffiffiffi Akþ2 2 p
Z
2
^f ðxÞðixÞk eð1þA4xÞ
þixz
dx;
0
ð3:1 Þ
R
where A ¼ a þ bi; z ¼ x þ yi; a; b; x; y 2 R; a > 0, and wðtÞ is the wavelet function in Proposition 3.1. The reproducing kernel of the image space Hk of the above formula is
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S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Z
Z 1 t z 0 12 t z0 1 12 0 1 ^ xÞwðA ^ 0 xÞeixðzz0 Þ dx 2 wðA ¼ K A; z; A0 ; z0 ¼ A 2 w A w A A A 2p A0 R R 0
ð3:2Þ
0
where A0 ¼ a0 þ b i; z ¼ x0 þ y0 i, a0 ; b ; x0 ; y0 2 R, and a0 > 0. Then, we can obtain the theorem below. Theorem 3.2. The reproducing kernel function for the image space of the wavelet transform (3.1) is given by 2 !12 ½AþA0 þ2ðZZ0 Þi
p 1 1þ kþ 2 A2 þ A0 2 2 0 2 0 0 4ðA þA 2 Þ 0 K A; z; A ; z ¼ ðAA Þ e 8 4 8 9
2ðknÞ !2k > > k = A þ A0 þ 2iðz z0 Þ X pffiffiffiffi< A þ A0 þ 2iðz z0 Þ 2l n p þ C ð2n 1Þ!!2
2k 2kn > > : ; A2 þ A0 2 n¼1 A2 þ A0 2
ð3:3Þ
where, k 2 Zþ . Proof 2
AþA0 þ2ðZZ 0 Þi Z 1 ½ ðA2 þA0 2 Þ
p kþ12 2þ 4 A2 þA0 2 0 0 ð Þ K A; z; A ; z0 ¼ ðAA Þ e x2k e 8 R
2
xþ
AþA0 þ2iðzz0 Þ A2 þA0 2 4
2
¼
p 8
1 1þ 2 0 kþ2
ðAA Þ
e
½AþA0 þ2ðZZ0 Þi Z 4ðA2 þA0 2 Þ
A þ A0 þ 2iðz z0 Þ
x
dx !2k
A2 þA0 2 4
e
A2 þ A0 2
R
dx:
We denote
Z Ik A; z; A0 ; z0 ¼
x
A þ A0 þ 2iðz z0 Þ 2
A þA
R
!2k eð
02
A2 þA0 2 Þx2 4
dx:
We use
2 !2kl 3 2k
Z X 2 02 A þ A0 þ 2iðz z0 Þ 0 l l Þx2 0 4 5eðA þA 4 Ik A; z; A ; z ¼ C 2k x dx 2 02 R A þA l¼0 !2kl Z 2k X A þ A0 þ 2iðz z0 Þ A2 þA0 2 2 ¼ ð1Þl C l2k xl eð 4 Þx dx 2 02 R þ A A l¼0 where
Z
2 þA0 2 Þx2 4
l ðA
xe
dx ¼
R
A2 þ A0 2 4
!lþ1 2 Z
þ1
l x2
xe
dx þ ð1Þ
0
l
Z
þ1
l x2
xe
0
1 A2 þ A0 2 dx ¼ 2 4
!lþ1 2
h
i l þ 1 : 1 þ ð1Þl C 2
Furthermore, we have 2k
1X A þ A0 þ 2iðz z0 Þ Ik A; z; A0 ; z0 ¼ ð1Þl C l2k 2 l¼0 A2 þ A0 2
!2kl
A2 þ A0 2 4
!lþ1 2
h
i l þ 1 : 1 þ ð1Þl C 2
If l ¼ 2n þ 1; n ¼ 0; 1; 2; . . . ; k 1, then 1 þ ð1Þl ¼ 0, if l ¼ 2n; n ¼ 0; 1; 2; . . . ; k. Accordingly, 1 þ ð1Þl ¼ 2; thus, we can obtain
Ik A; z; A0 ; z0 ¼
2ðknÞ !12 !2k k A þ A0 þ 2iðz z0 Þ X pffiffiffiffi 1 A2 þ A0 2 A þ A0 þ 2iðz z0 Þ 2n 2n ¼ 2 C 2k C nþ p
2kn 2 02 2 4 A þA n¼0 A2 þ A0 2
2ðknÞ !12 k A þ A0 þ 2iðz z0 Þ X pffiffiffiffi A2 þ A0 2 2n C 2k ð2n 1Þ!!2n p: þ
2kn 4 n¼1 A2 þ A0 2 A2 þ A0 2 4
!12
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S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Then, we have
8 !2k pffiffiffiffi< A þ A0 þ 2iðz z0 Þ p : A2 þ A0 2 9
2ðknÞ > k = A þ A0 þ 2iðz z0 Þ X ð2n 1Þ!!2n : þ C 2l
2kn 2k > ; n¼1 A2 þ A0 2
½AþA0 þ2ðZZ0 Þi
p 1 1þ 2 0 0 kþ2 4ðA2 þA0 2 Þ 0 K A; z; A ; z ¼ ðAA Þ e 8
2
A2 þ A0 2 4
!12
h Lemma 3.1
8mð2m 1Þ D2m
8m þ 2
x2m2 þ
D2mþ1
x2m þ
m X 8ðm lÞð2m 2n 1Þ 2ðmlÞ2 n 2 þ 8ðm nÞ 2ðmlÞ C 2l ð2l 1Þ!!2 x þ x 2m D2mlþ1 D2ml l¼1
m 2ðmlÞþ2 X 2l1 nx ðC 2l1 2mþ1 þ C 2m Þð2l 1Þ!!2 D2mlþ2 l¼1
¼ ð2m þ 1Þ!!
1 Dmþ1
2mþ1 ;
ð3:4Þ
where D – 0; D 2 C. Proof. Given that m X
l C 2l 2m ð2l 1Þ!!2
8ðm lÞð2m 2l 1Þ D2ml
l¼m1
2ðC 12mþ1 þ C 12m Þð2 1Þ!!
x2m D
2mþ1
mþ1 x2ðml1Þ þ C 2m 2m ð2m 1Þ!!2
¼ ð2m þ 1Þ!!
1 mþ1
D
1 Dmþ1
þ
8m þ 2 D2mþ1
x2m
2mþ1 ;
as long as the following equation is established, then the Lemma is established.
8mð2m 1Þ 2m
x2m2 þ
m2 X
l C 2l 2m ð2l 1Þ!!2
8ðm nÞð2m 2n 1Þ
D l¼1 m X x2ðmlÞþ2 2l1 2l1 ðC 2mþ1 þ C 2m Þð2l 1Þ!!2l 2mlþ2 ¼ 0: D l¼2
D
2ml
x2ðml1Þ þ
m1 X
l C 2l 2m ð2l 1Þ!!2
2 þ 8ðm nÞ
l¼1
D2mlþ1
m1 X
2 þ 8ðm nÞ
x2ðmlÞ
In fact
8mð2m 1Þ
m2 X
x2m2 þ
l C 2l 2m ð2l 1Þ!!2
8ðm nÞð2m 2n 1Þ
l C 2l 2m ð2l 1Þ!!2 D D2ml l¼1 l¼1 m X x2ðmlÞþ2 8mð2m 1Þ 2m2 2 þ 8ðm 1Þ 2m2 2l1 2l1 ðC 2mþ1 þ C 2m Þð2l 1Þ!!2l 2mlþ2 ¼ x þ 2C 22m x 2m D D2m D l¼2 2m
ðC 32mþ1 þ C 32m Þ3!!22 þ
m 1 X
l C 2l 2m ð2l 1Þ!!2
x2m2
þ
2m2
D
m1 X
l1 C 2l2 2m ð2l 3Þ!!2
2mlþ1
D
l¼2
D2mlþ1
x2ðmlÞ
8ðm l þ 1Þð2m 2l þ 1Þx2ðmlÞ
l¼2
2 þ 8ðm lÞx2ðmlÞ
x2ðml1Þ þ
D2mlþ1
m 1 X
x2ðmlÞ
l¼2
D2mlþ1
2l2 lþ1 ðC 2l2 2m þ C 2m Þð2l þ 1Þ!!2
:
The coefficient of x2m2 is
x2ðmlÞ h D
2m
i 8mð2m 1Þ þ 2C 22m ð8m 6Þ 4ðC 32mþ1 þ C 32m Þ3!! ¼ 0:
When l P 2, the coefficient of x2m2l is m 2 X
x2ðmlÞ
l¼1
D2mlþ1
ð2l 3Þ!!2lþ1
So the Lemma is established.
n o 2l 2lþ1 2lþ1 2C 2l2 2m ðm l þ 1Þð2m 2l þ 1Þ þ C 2m ð2l 1Þð1 þ 4m 4lÞ ðC 2mþ1 þ C 2m Þð2l þ 1Þð2l 1Þ ¼ 0:
h
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S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Theorem 3.3. The expression of (3.3) of reproducing kernel function K A; z; A0 ; z0 for the image space of the wavelet transform (3.1) can be further written as 2 0
p kþ12 pffiffiffiffi A þ A 2 K A; z; A0 ; z0 ¼ ðAA0 Þ p 8 4
2
!12
½AþA0 þ2ðZZ0 Þi 4ðA2 þA0 2 Þ ;
12þ
@ 2k @zk @Z 0 k
e
ð3:5Þ
where A; A0 2 M p4 ¼ AjjargAj < p4 ; z; z0 2 C; k 2 Zþ . Proof. In fact
@ 2k @zk @Z 0 k
e
12þ
9
2ðknÞ 2 28 !2k ½AþA0 þ2ðZZ0 Þi ½AþA0 þ2ðZZ0 Þi < > > 0 k = 12þ A þ A0 þ 2iðz z0 Þ 0Þ X A þ A þ 2iðz z 2 0 2 0 2 2 4ðA þA Þ 4ðA þA Þ þ C 2l ð2n 1Þ!!2n : ¼e
2k 2kn 2 0 > > : ; A þA2 n¼1 A2 þ A0 2
It can be proved by mathematical induction. When k ¼ 1, the result is clearly established. We suppose that the equation is also established, when k ¼ m, that is
@
2m
@zm @Z 0 m
9
2ðmnÞ 2 28 !2m 0 ½AþA0 þ2ðZZ0 Þi ½AþA0 þ2ðZZ0 Þi > > 0Þ 0 m < = 12þ A þ A þ 2iðz z 0 X A þ A þ 2iðz z Þ 2l n 4ðA2 þA0 2 Þ 4ðA2 þA0 2 Þ þ C ð2n 1Þ!!2 : ¼e
2m 2mn 2 02 > > : ; A þA n¼1 A2 þ A0 2
12þ
e
Next, we prove the equation is established, when k ¼ m þ 1, that is
@ 2mþ2 @zmþ1 @Z 0 mþ1
12þ
e
2 28 !2mþ2 ½AþA0 þ2ðZZ0 Þi ½AþA0 þ2ðZZ0 Þi < 12þ A þ A0 þ 2iðz z0 Þ 2 0 2 0 2 2 4ðA þA Þ 4ðA þA Þ þ ¼e : A2 þ A0 2
2 þ 8m 2
0 2 2mþ1
½A þ A0 þ 2ðZ Z 0 Þi
ðA þ A Þ 8h i2m2nþ2 2 9 0 0 ½AþA þ2ðZZ Þi m > = < A þ A0 þ 2iðz z0 Þ h i 12þ X 2m2 8mð2m 1Þ 0 2l n 4ðA2 þA0 2 Þ 0 þ A þ A þ 2ðZ Z Þi þe C 2m ð2n 1Þ!!2
2mnþ2 2m ; > : n¼1 ðA2 þ A0 2 Þ A2 þ A0 2 9 = i2m2n 8ðm nÞð2m 2n 1Þ h i2m2n2 > 2 þ 8ðm nÞ h 0 0 0 0 þ A þ A þ 2ðZ Z Þi : þ
2mnþ1 A þ A þ 2ðZ Z Þi 2mn > ; ðA2 þ A0 2 Þ A2 þ A0 2
2m
ð3:6Þ
We denote that D2 ¼ A2 þ A0 2 ; x ¼ A þ A0 þ 2iðz z0 Þ. The above equation can be simplified to 2
@ 2mþ2 @zmþ1 @Z 0 mþ1
12þ
e
½AþA0 þ2ðZZ0 Þi 2mþ2 m 2 2 X x 2 þ 8m 8mð2m 1Þ 2m2 1þ x 1þ x 4ðA2 þA0 2 Þ þ e 2 4D2 ¼ e 2 4D2 þ 2mþ1 x2m þ x C 2l 2m ð2n 2mþ2 2m D D D n¼1 2m2nþ2 x 2 þ 8ðm nÞ 2m2n 8ðm nÞð2m 2n 1Þ 2m2n2 : þ x þ x 1Þ!!2n D2mnþ1 D2mn D2mnþ2
2n1 2n1 As C 2n 2m ¼ C 2mþ1 þ C 2m , then we have
@ 2mþ2 @zmþ1 @Z 0 mþ1
12þ
e
½AþA0 þ2ðZZ 0 Þi 4ðA2 þA0 2 Þ
2
12þ x
¼e
2
4D2
x2mþ2
þe
12þ
x2 4D2
2 þ 8m
x2m þ
8mð2m 1Þ
x2m2
D2mþ1 D2m D2mþ2 m X 8ðm nÞð2m 2n 1Þ 2m2n2 n 2 þ 8ðm nÞ 2m2n þe C 2l ð2n 1Þ!!2 x þ x 2m D2mnþ1 D2mn n¼1 m m 2m2nþ2 2m2nþ2 2 X 2 X 1þ x 1þ x nx 2n1 nx C 2n e 2 4D2 ðC 2mþ1 þ C 2n1 : þ e 2 4D2 2mþ2 ð2n 1Þ!!2 2m Þð2n 1Þ!!2 2mnþ2 D D2mnþ2 n¼1 n¼1 2 12þ x 2 4D
We can obtain by Lemma 3.1.
@ 2mþ2
12þ
e
½AþA0 þ2ðZZ 0 Þi 4ðA2 þA0 2 Þ
2
¼e
12þ
x2 4D2
x2mþ2
2 4D2
12þ x
þe
2 þ 8m
x2m þ
8mð2m 1Þ
x2m2
D2mþ1 D2m D2mþ2 @zmþ1 @Z 0 mþ1 m 2 X 8ðm nÞð2m 2n 1Þ 2m2n2 1þ x n 2 þ 8ðm nÞ 2m2n C 2l x þ x þ e 2 4D2 2m ð2n 1Þ!!2 2mnþ1 2mn D D n¼1 m m 2m2nþ2 2 2 X X x2m2nþ2 1þ x 1þ x nx 2n1 2n1 þ e 2 4D2 C 2n e 2 4D2 ðC 2mþ1 þ C 2m Þð2n 1Þ!!2n 2mnþ2 2mþ2 ð2n 1Þ!!2 2mnþ2 D D n¼1 n¼1
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
(
m 2m2nþ2 X 1 nx þ mþ1 ð2m þ 1Þ!!2mþ1 þ C 2n 2mþ2 ð2n 1Þ!!2 2mþ2 D D D2mnþ2 n¼1 ( ) 2 X 2n x2mþ2 mþ1 x2m2nþ2 1þ x þ C 2mþ2 ð2n 1Þ!!2n 2mnþ2 : ¼ e 2 4D2 2mþ2 D D n¼1 12þ x
¼e
2
4D2
x2mþ2
123
)
Eq. (3.6) has been proved. That is, Eq. (3.5) is established 2 0
p kþ12 pffiffiffiffi A þ A 2 K A; z; A0 ; z0 ¼ ðAA0 Þ p 8 4
!12
2
@ 2k @zk @Z 0 k
where A; A0 2 M p4 ¼ AjjargAj < p4 ; z; z0 2 C; k 2 Zþ .
Eq. (3.5) implies that K A; z; A0 ; z0
e
12þ
½AþA0 þ2ðZZ0 Þi 4ðA2 þA0 2 Þ ;
h
is analytic on M p4 C as a function in ðA; zÞ, and it is anti-analytic on M p4 C as a
function in ðA0 ; z0 Þ. For the function f 2 L2 ðRÞ, in particular, we find that the image ðT wav f Þða; xÞ can be extended analytically onto the space M p4 C with the form ðT wav f ÞðA; zÞ, and the image ðT wav f Þða; xÞ can be characterised as the members of the Hilbert space Hk admitting the reproducing kernel function (3.3). 4. The reproducing kernel space when the scale factor is fixed Now, we merely consider the wavelet transform for any fixed A 2 M p4 . Then, we obtain the corresponding reproducing 0 kernel by setting A ¼ A 2
pffiffiffiffiffiffiffi ½aþðZZ0 Þi
12þ 1 1 p 2p 2 @ 2k 2 kþ2 2 2 2 0 0 2ða2 b2 Þ 0 K A; z; A ; z ¼ K A z; z ¼ ða þ b Þ ða b Þ e : 8 @zk @Z 0 k
For any fixed A 2 M p4 , and for any f 2 L2 ðRÞ, the image ðT wav f Þða; xÞ of the wavelet transform (3.1) can be characterised by the members of Hilbert space HkA admitting the above reproducing kernel and composed with all entire functions. Theorem 4.1. Let
pffiffiffiffiffiffiffi 2 0 1 1 1 ½aþðZZ Þi p 2p 2 2 kþ 2 þ K F ðkÞ z; z0 ¼ ða þ b Þ 2 ða2 b Þ 2 e 2 2ða2 b2 Þ ; 8
k 2 Zþ :
The function K F ðkÞ z; z0 is the reproducing kernel of Hilbert space HF ðkÞ composing all entire function hðzÞ with finite norms
khðzÞkF ðkÞ
8 > <
91 2 ( ) pffiffiffi > ZZ = a 2 4 2 1 2ðy 2Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy ¼ jhðzÞj exp 2 > > 2 a2 b :p2 pða2 b2 Þða2 þ b2 Þ2kþ1 C ;
and, we have the isometric identity ðkÞ
kh ðzÞkHK ¼ khkHF ðkÞ : A
Proof. Let
sðzÞ ¼
8 91
: 32ða2 b Þ ;
2 þ4azi2z2 4ða2 b2 Þ
14þa
e
and then, we have
pffiffiffiffiffiffiffi 2 0 1 1 1 ½aþðZZ Þi p 2p 2 2 kþ 2 þ K F ðkÞ z; z0 ¼ sðzÞsðz0 ÞK z; z0 ¼ ða þ b Þ 2 ða2 b Þ 2 e 2 2ða2 b2 Þ ; 8 where
K z; z0 ¼
zz0 a2
b
2
:
By Lemma 2.1 and Definition 2.3, we find that the function K F ðkÞ z; z0 is the reproducing kernel of the Hilbert space HF ðkÞ with the finite norm
124
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
( ( ) )12 2 ZZ 2 hðzÞ 1 ¼ hðzÞ exp jzj dxdy khðzÞkF ðkÞ ¼ sðzÞ 2 pða2 b2 Þ C sðzÞ a2 b Hk By the Euler formula
eðaþibÞx ¼ eax ðcosbx þ isinbxÞ and
jsðzÞj2 ¼ jsðzÞ2 j: we can obtain
khðzÞkF ðkÞ
8 > <
91 2 ( pffiffiffi ) > ZZ = a 2 4 2 1 2 y2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy ¼ jhðzÞj2 exp : 2 > > 2 a2 b :p2 pða2 b2 Þða2 þ b2 Þ2kþ1 C ;
Then, we have the isometric identity ðkÞ
kh ðzÞkHK ¼ khkHF ðkÞ :
A
Note that for any fixed A 2 M p4
1 tz ;z2C A2 w A
is complete in L2 ðRÞ. Then, we have the following characterisation of ðT wav f ÞA ðzÞ. Theorem 4.2. For any fixed A 2 M p4 ; k 2 Zþ , the image ðT wav f ÞA ðzÞ of a family of complex wavelet transform for f 2 L2 ðRÞ are characterised with properties that ðT wav f ÞA ðzÞ are entire functions that satisfy the following formula
ZZ
jðT wav f ÞA ðzÞj2 exp ðkÞ
C
( 2 ) 1 2 y 2a dxdy < 1: 2 2 a2 b
Furthermore, then we obtain the isometrical identity
kf k2 ¼
( pffiffiffi 2 ) ZZ 4 2 1 2 y 2a ðkÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy: jðT wav f ÞA ðzÞj2 exp 2 2kþ1 2 C a2 b p2 pða2 b2 Þða2 þ b2 Þ
ð4:1Þ
Proposition 4.1. The images f ðpÞ of the linear transform (2.1) for H form precisely the functional Hilbert space Hk admitting the reproducing kernel kðp; qÞ defined by (2.2), which is uniquely determined by the reproducing kernel kðp; qÞ. Then, we have the inequality
kf kHk 6 kfkH : Furthermore, for any f 2 Hk , there exists a uniquely determined member f of H such that
f ðpÞ ¼ ðf ; hðpÞÞH and
kf kHk ¼ kf kH : We shall assume that the norms in H and Hk are realised in terms of r-finite positive measures dm and dl on measurable spaces T and E in the following manners
H ¼ L2 ðT; dmÞ;
Hk L2 ðE; dlÞ:
Thus, we shall consider the integral transform
f ðpÞ ¼
Z
FðtÞhðt; pÞdmðtÞ;
T
where hðt; pÞ is a complex-valued function on T E with
hð; pÞ 2 L2 ðT; dmÞ
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
125
and for the functions F satisfying
F 2 L2 ðT; dmÞ; the corresponding reproducing kernel is
kðp; qÞ ¼
Z
hðt; qÞhðt; pÞdmðtÞ: T
Proposition 4.2. Suppose that there exists an exhaustion fEN g1 N¼1 of dl-measurable sets of E satisfying (1) E1 E2 , (2) [1 E ¼ E, R N¼1 N (3) EN Kðp; qÞdlðpÞ < 1. For a member f of Hk , if
Z
f ðpÞhðt; pÞdlðpÞ 2 L2 ðT; dmÞ;
EN
we have the sequence
Z
f ðpÞhðt; pÞdlðpÞ
1
EN
m; N¼1
that converges strongly to F in Proposition 4.1. That is
F ðtÞ ¼ s lim
Z
N!1
f ðpÞhðt; pÞdlðpÞ
EN
for the function F satisfying
f ðpÞ ¼
Z
F ðtÞhðt; pÞdlðtÞ
T
and
kf kHk ¼ kF kL2ðT; dlÞ : Next, we shall derive a characterisation of the images ðT wav f ÞA ðxÞ as real variable functions. By the Plancherel theorem, the isometrical identity (4.3) and Proposition 4.1, we have Theorem 4.3. For any fixed A 2 M p4 ; k 2 Zþ , and for f 2 L2 ðRÞ, the image ðT wav f ÞA ðxÞ of (3.1) are characterised by the properties 1 that are of class C ðRÞ. Then, we furthermore obtain the isometrical identity
kf k2 ¼
2 pffiffiffi 2 n Z nk wav 1 X 4 2 2n ða2 b Þ @ ðT f ÞðxÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx; nk 2kþ1 @x n! R n¼0 p2 pða2 þ b2 Þ
ð4:4Þ
where
@ 1 ðT wav f ÞðxÞ ¼ @x1
Z
x
ðT wav f ÞðlÞdl
infty
and
! @ 2 @ 1 @ 1 ¼ : @x2 @x1 @x1 From the identities (4.3) and (4.4), we obtain the family of the complex wavelet transform inversion formulas for (3.1) by Proposition 4.1, in a general argument. Theorem 4.4. In the complex wavelet transform (3.1), for any fixed A 2 M p4 ; k 2 Zþ , we have the family of the complex wavelet transform inversion formulas
pffiffiffiffiffiffi1 1 ðkÞ 2ðyaÞ2 ZZ 2 4 2e2 A2 tz ðkÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT wav f Þ ðzÞw e a2 b2 dxdy: 2kþ1 N!1 A EN p2 pða2 b2 Þða2 þ b2 Þ
f ¼ lim
ð4:5Þ
126
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
For any compact exhaustion fEN g1 N¼1 of C, and
pffiffiffiffiffiffi1 1 ! 2 n Z N X 4 2e2 A2 2n ða2 b Þ @ nk ðT wav f ÞðxÞ @ nk tx dx; f ¼ lim qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2kþ1 N!1 @xnk n! @xnk A R n¼0 p2 pða2 þ b2 Þ
ð4:6Þ
in the sense of strong convergence in L2 ðRÞ. 5. Conclusions In this paper, an admitting complex wavelet is constructed using a convolution formula, and the image space of the complex wavelet transform is characterised by the reproducing kernel throry. The results of Ref.[18] have been popularised and supported. Accordingly, this article should serve as the basis for additional research and further useful results. Acknowledgments The work is supported by a Grant from the Youth Natural Science Foundation of Heilongjiang Province, No. QC2011C103. References [1] X.W. Zhang, J.F. He, X.B. Zhang, The parameter estimation algorithm research for weak sinusoidal signal, J. Numer. Methods Comput. Appl. 3 (2012) 222–229. [2] L.Q. Li, Y.Y. Tang, Wavelet–Houch transform with applications in edge and target detections, Int. J. Wavelets Multiresolution Inf. Process. 4 (1992) 567– 587. [3] ZH.X. Cheng, SH.ZH. Yang, X.X. Feng, Wavelet Analysis Theory. The Algorithm, Progress and Application, The Defense Industry Press, Beijing, China, 2007. 05: 145–150. [4] F.Z. Geng, A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method, Appl. Math. Comput. 218 (8) (2011) 4211–4215. [5] N. Hauyashi, S. Saitoh, Analyticity and smoothing effect for the Schrodinger equation, Ann. Inst. Henripoincare 52 (1990) 163–173. [6] B.Y. Wu, X.Y. Li, Application of reproducing kernel method to third order three-point boundary value problems, Appl. Math. Comput. 217 (7) (2010) 3425–3428. [7] H. Long, X.J. Zhang, Construction and calculation of reproducing kernel determined by various linear differential operators, Appl. Math. Comput. 215 (2) (2009) 759–766. [8] M. Mouattamid, Recursive producing kernels Hilbert space using the theory of power kernels, Anal. Theory Appl. 2 (2012) 111–124. [9] J.S. Chen, C. Pan, H.P. Wang, A lagrangian reproducing kernel particle method for metal forming analysis, Comput. Mech. 6 (1998) 289–307. [10] N.R. Aluru, A reproducing kernel particle method for meshless analysis of microelectro mechanical systems, Comput. Mech. 23 (1999) 324–338. [11] N.R. Aluru, A point collocation method based on reproducing kernel approximations, Int. J. Numer. Methods Eng. 47 (2000) 1083–1121. [12] S. Saitoh, Reproducing kernels of the direct product of two Hilbert space, Riazi. J. Kar. Math. Assoc. 4 (1982) 1–20. [13] S. Saitoh, Representations of the norms in Bergman–Selberg spaces on strips and half planes, Complex Variables 19 (1992) 231–241. [14] S. Saitoh, Quadratic norm inequalities deduced from the theory of reproducing kernels, Linear Algebra Appl. 93 (5) (1987) 171–178. [15] L.J. Gu, C.X. Deng, Q.Y. Ling, Expressing wavelet transform with reproducing kernel, J. Math. 28 (2008) 507–513. [16] C.X. Deng, Q.Y. Ling, L.J. Gu, Characterization of image space of a wavelet transform, Int. J. Wavelets Multiresolution Inf. Process. 4 (2006) 547–557. [17] Q.Y. Ling, C.X. Deng, J. Hou, Characterization of image space of Gauss wavelet transform, ACTA Math. Sinica Chin. Ser. 51 (2008) 225–234. [18] S.S. Li, C.X. Deng, Z.X. Fu, The isometric and the inversion formulas of image space of Cgau wavelet transform, Chin. Ann. Math. 32 (2011) 121–128. [19] L.P. Castro, E.M. Rojas, S. Saitoh, Inversion from different kinds of information and real inversion formulas of the Laplace transform from a finite number of data, Math. Eng. Sci. Aerosp. 02 (2010) 181–190. [20] L.P. Castro, H. Fujiwara, S. Saitoh, Y. Sawano, A. Yamada, M. Yamada, Fundamental error estimates inequalities for the Tikhonov regularization using reproducing kernels, Int. Ser. Numer. Math. Inequalities Appl. 02 (2012) 87–101. [21] I. Daubechies, Ten lectures on wavelets, Soc. Ind. Appl. Math. 1 (1992) 256–263.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The isometric identities and inversion formulas of complex continuous wavelet transforms Shasha Li a,b,⇑, Caixia Deng c, Wei Sun a,c a
Institute of Mathematics, Jilin University, Changchun 130012, China Department of Mathematics, Daqing Normal University, Daqing 163712, China c School of Applied Sciences, Harbin University of Science and Technology, Harbin 150080, China b
a r t i c l e Keywords: Reproducing kernel Wavelet transform Image space
i n f o
a b s t r a c t The reproducing kernel function of the image space of a family of complex wavelet transforms is presented. An admissible wavelet is obtained by convolution computation. Next, the correlative characterisation of the image space of the family of complex wavelet transforms is provided when the scale is fixed. Furthermore, the isometric identities and inversion formulas are obtained, which provide a theoretic basis for investigating the image space of the general wavelet transform. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Wavelet analysis has been rapidly developed in the mathematical field. This analysis is a brilliant combination of pure mathematics and applied mathematics following the Fourier transform, which has been called a mathematics microscope. In addition, wavelet analysis is also one of the most obvious achievements of harmonic analysis in the past half century. Wavelet analysis can effectively provide useful information from signals, and it also can solve many difficult problems that cannot be solved using the Fourier transform. Currently, this analysis has made most of the important valuable achievements in, for instance, the field of signal analysis (see Ref. [1]), image processing (see Ref. [2]), sound processing (see Ref. [3]), and solutions of equations (see Refs. [4–7]). We all know that there is redundancy of information with wavelet transforms, and the wavelet transform coefficient has relevance in the wavelet transform image plane. We can also see that the reproducing the kernel Hilbert space is the basis of the wavelet transform, and the relevance region’s magnitude is given by the reproducing the kernel. In addition, we can prove that its magnitude will decrease if the scale decreases. Therefore, we can conclude that the reproducing the kernel space plays an important role in the reconstruction of the continuous wavelet transform (see Refs. [8–11]). With the development of wavelet analysis, reproducing kernel theory has attracted increasing attention from many scholars. For example, Saitoh described reproducing kernels of the direct product of two Hilbert spaces (see Refs. [12–14]). Deng and Du have described the image space of wavelet transform such as that of Shannon, Meyer and Littlewood-Paley (see Refs. [15–17]). Li et al. further studied the image space of the complex Gauss wavelet transform (see Ref. [18]). Castro et al. have identified a general discretisation method for solving wide classes of mathematical problems by applying the theory of reproducing kernels (see Refs. [19,20]). It can be observed that continuous wavelet transform is the foundation of the reproducing kernel function of the image space. Therefore, we can choose the most suitable wavelet bases according to the structure of the reproducing kernel, which is based on the information redundancy of the continuous wavelet transform. In this paper, we present a family of admissible complex wavelets and describe the spaces with the help of ⇑ Corresponding author at: Institute of Mathematics, Jilin University, Changchun 130012, China. E-mail addresses: [email protected] (S. Li), [email protected] (C. Deng), [email protected] (W. Sun). 0096-3003/$ - see front matter Ó 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.079
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
117
Fock kernel. The paper demonstrates isometry and inversion formulas under the fixed scale-factor conditions and further provides a theoretical foundation for the studies of the general wavelet transform. It is obvious that the results of Ref. [18] are a special case for this article. 2. Definition and properties In this section, we introduce the concept and properties involved with reproducing kernels and wavelets. Definition 2.1. Let H be a Hilbert function space. Its elements are real-valued or complex-valued functions on an abstract set E. The inner can be denoted by
hf ; gi ¼ hf ðÞ; gðÞi;
f; g 2 H
for any fixed q 2 E; Kðp; qÞ 2 H as a function in p, and for any f 2 H, and q 2 E, we have
f ðqÞ ¼ hf ðpÞ; Kðp; qÞi; then Kðp; qÞ will be called the reproducing kernel of the Hilbert function space H; H will be called the reproducing kernel Hilbert space (RKHS). Furthermore, the function Kðp; qÞ is uniquely determined by RKHS H. Therefore, it can be written as HK , that is, H ¼ HK . Furthermore, we can also write the Definition below for the reproducing kernel via linear transformation. Proposition 2.1. Let F ðEÞ be a linear space comprising all complex-valued functions on an abstract set E. Let H be a Hilbert space equipped with product h:; :iH . Let h; E ! H be a Hilbert space H-valued function on E. We consider the linear mapping which is L from Hinto F ðEÞas defined by
f ðpÞ ¼ ðLfÞðpÞ ¼ ðf; hðpÞÞH :
ð2:1Þ
We suppose that
kðp; qÞ ¼ hhðqÞ; hðpÞiH :
ð2:2Þ
Let RðLÞ be the range of L for H and we introduce the inner product in RðLÞ induced form the norm
kf kRðLÞ ¼ inf kfkH ;
f ¼ Lf:
Definition 2.2. For the Hilbert space F ðkÞ ðk > 0Þ comprising all entire functions f ðzÞ with finite norms
ZZ k
12 2 jf ðzÞj2 ekjzj dxdy < 1;
p
C
the function ekuz is the reproducing kernel. This space will be called the Fock space (or the Bargmann–Fock space; see Ref. [14]). Lemma 2.1. For a RKHS HK on E and for any non-vanishing complex-valued function sðpÞ on E,
ks ðp; qÞ ¼ sðpÞsðqÞkðp; qÞ;
p; q 2 E
is a reproducing kernel for the Hilbert space Hks , comprising all the functions fs ðpÞ on E that are expressible in the form
fs ðpÞ ¼ f ðpÞsðpÞ;
f 2 HK
and that is equipped with the inner product
hfs ; g s iHk ¼ s
fs g s ; ; s s HK
(see Ref. [12]). ^ xÞ satisfies the admissible condition Definition 2.3. Let wðtÞ 2 L2 ðRÞ , and its Fourier transform wð
Cw ¼
Z R
^ xÞj2 jwð dx < 1; j xj
then wðtÞ is called the basic wavelet or mother wavelet, and it is also known as an admissible wavelet.
118
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Definition 2.4. Let f ðtÞ 2 L2 ðRÞ; wðtÞ be a basic wavelet, and
1 tb ; wa; b ðtÞ ¼ pffiffiffiffiffiffi w a jaj where a; b 2 R; a – 0, then the function wa; b ðtÞ is a continuous wavelet that is generated by wðtÞ and is related to with parameters a and b, and
1 ðT wav f Þða; bÞ ¼ pffiffiffiffiffiffi jaj
tb dt f ðtÞw a R
Z
is the continuous wavelet transform of f ðtÞ. The continuous wavelet transform of f ðtÞ in a frequency domain is given by
ðT wav f Þða; bÞ ¼ hf ; wa; b i ¼
pffiffiffi Z a 1 ^ ^ a; b ^f ðxÞwða ^ xÞeibx dx: hf ; w i ¼ 2p R 2p
The continuous wavelet transform of f ðtÞ can also be written as a convolution type in the frequency domain as given by 1
ðT wav f Þða; bÞ ¼ jaj2 f wjaj ðbÞ
0
where
b : wjaj ðbÞ ¼ jaj1 w a Thus, a wavelet transform can be observed as a convolution computation of the signal and filter. wjaj ðbÞ can be comprehended as a high-pass filter from an engineering point of view. Lemma 2.2. For all f ðtÞ; gðtÞ 2 L2 ðRÞ, we have
hf ; gi ¼
1 Cw
Z Z R
ðT wav f Þða; bÞðT wav gÞða; bÞ R
1 dadb a2
where a; b 2 R; a – 0 (see Ref. [21]). Indeed, we have directly
Z Z R
ðT
wav
R
f Þða; bÞðT
wav
#" 1 Z # Z Z " 12 Z 2 jaj dadb 0 ib l 0 0 ^f ðlÞeibl wða 0 ^ lÞdl jaj ^ g^ðl Þe wðal Þdl a2 2 p 2 p R R R R Z Z Z da ^f ðlÞg^ðlÞjwða ^ lÞj2 dl ¼ C w f ðxÞgðxÞdx: ¼ 2p R jaj R R
dadb gÞða; bÞ 2 ¼ a
Thus, we find that the reproducing kernel of the image space of wavelet transform is given by
Kða0 ; b0 ; a; bÞ ¼ hwa;b ðxÞ; wa0 ; b0 ðxÞi ¼
Z
wa; b ðxÞwa0 ; b0 ðxÞdx:
R
The norm in the RKHS Hk is given by
1 Cw
Z Z R
jðT wav f Þða; bÞj2 R
12 1 dadb ; a2
2
where f 2 L ðRÞ. The finiteness of the above integral does not, however, characterise the members in Hk . That is, Hk forms a subspace in the Hilbert space comprising all functions on R R with above finite norms. Thus, we are interested in the characterisation of the subspace Hk . 3. The image space of a family of complex wavelet transform Theorem 3.1. Let wðtÞ be a basic wavelet, gðtÞ 2 L1 ðRÞ be a bounded function, and then its convolution w gðtÞ is also a basic wavelet. ^ðxÞ is bounded. By the Cauchy–Schwarz inequality, we have Proof. As gðtÞ 2 L1 ðRÞ; g
119
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Z Z w gðtÞj2 dt ¼
Z 2 2 Z Z wðt uÞgðuÞdu dt 6 jwðt uÞjjgðuÞjdu dt R R R R Z Z Z 6 jwðt uÞj2 jgðuÞjdu jgðuÞjdu dt
R
¼
R
R
R
R
Z Z
R
Z 2 Z Z jwðt uÞj2 dtjgðuÞj du jgðuÞjdu ¼ jwðtÞj2 dt jgðuÞjdu R
R
R
< þ1: Then, we can obtain w gðtÞ 2 L2 ðRÞ and
Z R
jw ^ gðxÞj2 dx ¼ j xj
Z R
^ xÞj2 jg^ðxÞj2 jwð dx 6 supjg^ðxÞj2 jxj
Z R
^ xÞj2 jwð dx < þ1: j xj
By Definition 2.4, thus, the function w xðtÞ is an admissible wavelet function. 2
h 2
Proposition 3.1. Let f ðtÞ ¼ eit2t , and gðtÞ ¼ f ðk1Þ ðtÞ, where k 2 Zþ is a positive integer, and w1 ðtÞ ¼ eit2t ði 4tÞ, then the convolution function wðtÞ ¼ w1 gðtÞ is a basic wavelet. Proof. The Fourier transform of w1 ðtÞ is
^ 1 ð xÞ ¼ w
pffiffiffiffiffiffiffi ð1þxÞ2 2p ixe 8 : 2
The Fourier transform of f ðtÞ is
pffiffiffiffiffiffiffi 2p ð1þxÞ2 e 8 : 2
^f ðxÞ ¼
The Fourier transform of gðtÞ is
g^ðxÞ ¼
pffiffiffiffiffiffiffi ð1þxÞ2 2p k1 ðixÞ e 8 : 2
Then, we can find that w1 ðtÞ satisfies the admissible condition, gðtÞ 2 L1 ðRÞ, and gðtÞ is bounded. So, we can see by Theorem 3.1 that
w ¼ w1 gðtÞ is a basic wavelet function.
h
The Fourier transform of wðtÞ is
^ xÞ ¼ w ^ 1 ðxÞg^ðxÞ ¼ p ðixÞk eð1þ4xÞ : wð 2 2
Then, we can find the conclusion by the inversion formula of Fourier transform. When k ¼ 1; wðtÞ below is the cgau wavelet (complex Gauss wavelet) function
wðtÞ ¼
pffiffiffi 2 2eitt ði 2tÞ:
When k ¼ 2; wðtÞ below is the complex Mexihat wavelet function
pffiffiffiffi 2 wðtÞ ¼ 2 peitt ð4t 2 þ 4it 3Þ: By the symmetry in a, we merely consider the wavelet transform for a > 0. We shall consider the wavelet transform in the complex form below
ðT wav f ÞðA; zÞ ¼
1 tz dt; f ðtÞA2 w A R
Z
ð3:1Þ
or 1 1 ðT wav f ÞðA; zÞ ¼ pffiffiffiffi Akþ2 2 p
Z
2
^f ðxÞðixÞk eð1þA4xÞ
þixz
dx;
0
ð3:1 Þ
R
where A ¼ a þ bi; z ¼ x þ yi; a; b; x; y 2 R; a > 0, and wðtÞ is the wavelet function in Proposition 3.1. The reproducing kernel of the image space Hk of the above formula is
120
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Z
Z 1 t z 0 12 t z0 1 12 0 1 ^ xÞwðA ^ 0 xÞeixðzz0 Þ dx 2 wðA ¼ K A; z; A0 ; z0 ¼ A 2 w A w A A A 2p A0 R R 0
ð3:2Þ
0
where A0 ¼ a0 þ b i; z ¼ x0 þ y0 i, a0 ; b ; x0 ; y0 2 R, and a0 > 0. Then, we can obtain the theorem below. Theorem 3.2. The reproducing kernel function for the image space of the wavelet transform (3.1) is given by 2 !12 ½AþA0 þ2ðZZ0 Þi
p 1 1þ kþ 2 A2 þ A0 2 2 0 2 0 0 4ðA þA 2 Þ 0 K A; z; A ; z ¼ ðAA Þ e 8 4 8 9
2ðknÞ !2k > > k = A þ A0 þ 2iðz z0 Þ X pffiffiffiffi< A þ A0 þ 2iðz z0 Þ 2l n p þ C ð2n 1Þ!!2
2k 2kn > > : ; A2 þ A0 2 n¼1 A2 þ A0 2
ð3:3Þ
where, k 2 Zþ . Proof 2
AþA0 þ2ðZZ 0 Þi Z 1 ½ ðA2 þA0 2 Þ
p kþ12 2þ 4 A2 þA0 2 0 0 ð Þ K A; z; A ; z0 ¼ ðAA Þ e x2k e 8 R
2
xþ
AþA0 þ2iðzz0 Þ A2 þA0 2 4
2
¼
p 8
1 1þ 2 0 kþ2
ðAA Þ
e
½AþA0 þ2ðZZ0 Þi Z 4ðA2 þA0 2 Þ
A þ A0 þ 2iðz z0 Þ
x
dx !2k
A2 þA0 2 4
e
A2 þ A0 2
R
dx:
We denote
Z Ik A; z; A0 ; z0 ¼
x
A þ A0 þ 2iðz z0 Þ 2
A þA
R
!2k eð
02
A2 þA0 2 Þx2 4
dx:
We use
2 !2kl 3 2k
Z X 2 02 A þ A0 þ 2iðz z0 Þ 0 l l Þx2 0 4 5eðA þA 4 Ik A; z; A ; z ¼ C 2k x dx 2 02 R A þA l¼0 !2kl Z 2k X A þ A0 þ 2iðz z0 Þ A2 þA0 2 2 ¼ ð1Þl C l2k xl eð 4 Þx dx 2 02 R þ A A l¼0 where
Z
2 þA0 2 Þx2 4
l ðA
xe
dx ¼
R
A2 þ A0 2 4
!lþ1 2 Z
þ1
l x2
xe
dx þ ð1Þ
0
l
Z
þ1
l x2
xe
0
1 A2 þ A0 2 dx ¼ 2 4
!lþ1 2
h
i l þ 1 : 1 þ ð1Þl C 2
Furthermore, we have 2k
1X A þ A0 þ 2iðz z0 Þ Ik A; z; A0 ; z0 ¼ ð1Þl C l2k 2 l¼0 A2 þ A0 2
!2kl
A2 þ A0 2 4
!lþ1 2
h
i l þ 1 : 1 þ ð1Þl C 2
If l ¼ 2n þ 1; n ¼ 0; 1; 2; . . . ; k 1, then 1 þ ð1Þl ¼ 0, if l ¼ 2n; n ¼ 0; 1; 2; . . . ; k. Accordingly, 1 þ ð1Þl ¼ 2; thus, we can obtain
Ik A; z; A0 ; z0 ¼
2ðknÞ !12 !2k k A þ A0 þ 2iðz z0 Þ X pffiffiffiffi 1 A2 þ A0 2 A þ A0 þ 2iðz z0 Þ 2n 2n ¼ 2 C 2k C nþ p
2kn 2 02 2 4 A þA n¼0 A2 þ A0 2
2ðknÞ !12 k A þ A0 þ 2iðz z0 Þ X pffiffiffiffi A2 þ A0 2 2n C 2k ð2n 1Þ!!2n p: þ
2kn 4 n¼1 A2 þ A0 2 A2 þ A0 2 4
!12
121
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Then, we have
8 !2k pffiffiffiffi< A þ A0 þ 2iðz z0 Þ p : A2 þ A0 2 9
2ðknÞ > k = A þ A0 þ 2iðz z0 Þ X ð2n 1Þ!!2n : þ C 2l
2kn 2k > ; n¼1 A2 þ A0 2
½AþA0 þ2ðZZ0 Þi
p 1 1þ 2 0 0 kþ2 4ðA2 þA0 2 Þ 0 K A; z; A ; z ¼ ðAA Þ e 8
2
A2 þ A0 2 4
!12
h Lemma 3.1
8mð2m 1Þ D2m
8m þ 2
x2m2 þ
D2mþ1
x2m þ
m X 8ðm lÞð2m 2n 1Þ 2ðmlÞ2 n 2 þ 8ðm nÞ 2ðmlÞ C 2l ð2l 1Þ!!2 x þ x 2m D2mlþ1 D2ml l¼1
m 2ðmlÞþ2 X 2l1 nx ðC 2l1 2mþ1 þ C 2m Þð2l 1Þ!!2 D2mlþ2 l¼1
¼ ð2m þ 1Þ!!
1 Dmþ1
2mþ1 ;
ð3:4Þ
where D – 0; D 2 C. Proof. Given that m X
l C 2l 2m ð2l 1Þ!!2
8ðm lÞð2m 2l 1Þ D2ml
l¼m1
2ðC 12mþ1 þ C 12m Þð2 1Þ!!
x2m D
2mþ1
mþ1 x2ðml1Þ þ C 2m 2m ð2m 1Þ!!2
¼ ð2m þ 1Þ!!
1 mþ1
D
1 Dmþ1
þ
8m þ 2 D2mþ1
x2m
2mþ1 ;
as long as the following equation is established, then the Lemma is established.
8mð2m 1Þ 2m
x2m2 þ
m2 X
l C 2l 2m ð2l 1Þ!!2
8ðm nÞð2m 2n 1Þ
D l¼1 m X x2ðmlÞþ2 2l1 2l1 ðC 2mþ1 þ C 2m Þð2l 1Þ!!2l 2mlþ2 ¼ 0: D l¼2
D
2ml
x2ðml1Þ þ
m1 X
l C 2l 2m ð2l 1Þ!!2
2 þ 8ðm nÞ
l¼1
D2mlþ1
m1 X
2 þ 8ðm nÞ
x2ðmlÞ
In fact
8mð2m 1Þ
m2 X
x2m2 þ
l C 2l 2m ð2l 1Þ!!2
8ðm nÞð2m 2n 1Þ
l C 2l 2m ð2l 1Þ!!2 D D2ml l¼1 l¼1 m X x2ðmlÞþ2 8mð2m 1Þ 2m2 2 þ 8ðm 1Þ 2m2 2l1 2l1 ðC 2mþ1 þ C 2m Þð2l 1Þ!!2l 2mlþ2 ¼ x þ 2C 22m x 2m D D2m D l¼2 2m
ðC 32mþ1 þ C 32m Þ3!!22 þ
m 1 X
l C 2l 2m ð2l 1Þ!!2
x2m2
þ
2m2
D
m1 X
l1 C 2l2 2m ð2l 3Þ!!2
2mlþ1
D
l¼2
D2mlþ1
x2ðmlÞ
8ðm l þ 1Þð2m 2l þ 1Þx2ðmlÞ
l¼2
2 þ 8ðm lÞx2ðmlÞ
x2ðml1Þ þ
D2mlþ1
m 1 X
x2ðmlÞ
l¼2
D2mlþ1
2l2 lþ1 ðC 2l2 2m þ C 2m Þð2l þ 1Þ!!2
:
The coefficient of x2m2 is
x2ðmlÞ h D
2m
i 8mð2m 1Þ þ 2C 22m ð8m 6Þ 4ðC 32mþ1 þ C 32m Þ3!! ¼ 0:
When l P 2, the coefficient of x2m2l is m 2 X
x2ðmlÞ
l¼1
D2mlþ1
ð2l 3Þ!!2lþ1
So the Lemma is established.
n o 2l 2lþ1 2lþ1 2C 2l2 2m ðm l þ 1Þð2m 2l þ 1Þ þ C 2m ð2l 1Þð1 þ 4m 4lÞ ðC 2mþ1 þ C 2m Þð2l þ 1Þð2l 1Þ ¼ 0:
h
122
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
Theorem 3.3. The expression of (3.3) of reproducing kernel function K A; z; A0 ; z0 for the image space of the wavelet transform (3.1) can be further written as 2 0
p kþ12 pffiffiffiffi A þ A 2 K A; z; A0 ; z0 ¼ ðAA0 Þ p 8 4
2
!12
½AþA0 þ2ðZZ0 Þi 4ðA2 þA0 2 Þ ;
12þ
@ 2k @zk @Z 0 k
e
ð3:5Þ
where A; A0 2 M p4 ¼ AjjargAj < p4 ; z; z0 2 C; k 2 Zþ . Proof. In fact
@ 2k @zk @Z 0 k
e
12þ
9
2ðknÞ 2 28 !2k ½AþA0 þ2ðZZ0 Þi ½AþA0 þ2ðZZ0 Þi < > > 0 k = 12þ A þ A0 þ 2iðz z0 Þ 0Þ X A þ A þ 2iðz z 2 0 2 0 2 2 4ðA þA Þ 4ðA þA Þ þ C 2l ð2n 1Þ!!2n : ¼e
2k 2kn 2 0 > > : ; A þA2 n¼1 A2 þ A0 2
It can be proved by mathematical induction. When k ¼ 1, the result is clearly established. We suppose that the equation is also established, when k ¼ m, that is
@
2m
@zm @Z 0 m
9
2ðmnÞ 2 28 !2m 0 ½AþA0 þ2ðZZ0 Þi ½AþA0 þ2ðZZ0 Þi > > 0Þ 0 m < = 12þ A þ A þ 2iðz z 0 X A þ A þ 2iðz z Þ 2l n 4ðA2 þA0 2 Þ 4ðA2 þA0 2 Þ þ C ð2n 1Þ!!2 : ¼e
2m 2mn 2 02 > > : ; A þA n¼1 A2 þ A0 2
12þ
e
Next, we prove the equation is established, when k ¼ m þ 1, that is
@ 2mþ2 @zmþ1 @Z 0 mþ1
12þ
e
2 28 !2mþ2 ½AþA0 þ2ðZZ0 Þi ½AþA0 þ2ðZZ0 Þi < 12þ A þ A0 þ 2iðz z0 Þ 2 0 2 0 2 2 4ðA þA Þ 4ðA þA Þ þ ¼e : A2 þ A0 2
2 þ 8m 2
0 2 2mþ1
½A þ A0 þ 2ðZ Z 0 Þi
ðA þ A Þ 8h i2m2nþ2 2 9 0 0 ½AþA þ2ðZZ Þi m > = < A þ A0 þ 2iðz z0 Þ h i 12þ X 2m2 8mð2m 1Þ 0 2l n 4ðA2 þA0 2 Þ 0 þ A þ A þ 2ðZ Z Þi þe C 2m ð2n 1Þ!!2
2mnþ2 2m ; > : n¼1 ðA2 þ A0 2 Þ A2 þ A0 2 9 = i2m2n 8ðm nÞð2m 2n 1Þ h i2m2n2 > 2 þ 8ðm nÞ h 0 0 0 0 þ A þ A þ 2ðZ Z Þi : þ
2mnþ1 A þ A þ 2ðZ Z Þi 2mn > ; ðA2 þ A0 2 Þ A2 þ A0 2
2m
ð3:6Þ
We denote that D2 ¼ A2 þ A0 2 ; x ¼ A þ A0 þ 2iðz z0 Þ. The above equation can be simplified to 2
@ 2mþ2 @zmþ1 @Z 0 mþ1
12þ
e
½AþA0 þ2ðZZ0 Þi 2mþ2 m 2 2 X x 2 þ 8m 8mð2m 1Þ 2m2 1þ x 1þ x 4ðA2 þA0 2 Þ þ e 2 4D2 ¼ e 2 4D2 þ 2mþ1 x2m þ x C 2l 2m ð2n 2mþ2 2m D D D n¼1 2m2nþ2 x 2 þ 8ðm nÞ 2m2n 8ðm nÞð2m 2n 1Þ 2m2n2 : þ x þ x 1Þ!!2n D2mnþ1 D2mn D2mnþ2
2n1 2n1 As C 2n 2m ¼ C 2mþ1 þ C 2m , then we have
@ 2mþ2 @zmþ1 @Z 0 mþ1
12þ
e
½AþA0 þ2ðZZ 0 Þi 4ðA2 þA0 2 Þ
2
12þ x
¼e
2
4D2
x2mþ2
þe
12þ
x2 4D2
2 þ 8m
x2m þ
8mð2m 1Þ
x2m2
D2mþ1 D2m D2mþ2 m X 8ðm nÞð2m 2n 1Þ 2m2n2 n 2 þ 8ðm nÞ 2m2n þe C 2l ð2n 1Þ!!2 x þ x 2m D2mnþ1 D2mn n¼1 m m 2m2nþ2 2m2nþ2 2 X 2 X 1þ x 1þ x nx 2n1 nx C 2n e 2 4D2 ðC 2mþ1 þ C 2n1 : þ e 2 4D2 2mþ2 ð2n 1Þ!!2 2m Þð2n 1Þ!!2 2mnþ2 D D2mnþ2 n¼1 n¼1 2 12þ x 2 4D
We can obtain by Lemma 3.1.
@ 2mþ2
12þ
e
½AþA0 þ2ðZZ 0 Þi 4ðA2 þA0 2 Þ
2
¼e
12þ
x2 4D2
x2mþ2
2 4D2
12þ x
þe
2 þ 8m
x2m þ
8mð2m 1Þ
x2m2
D2mþ1 D2m D2mþ2 @zmþ1 @Z 0 mþ1 m 2 X 8ðm nÞð2m 2n 1Þ 2m2n2 1þ x n 2 þ 8ðm nÞ 2m2n C 2l x þ x þ e 2 4D2 2m ð2n 1Þ!!2 2mnþ1 2mn D D n¼1 m m 2m2nþ2 2 2 X X x2m2nþ2 1þ x 1þ x nx 2n1 2n1 þ e 2 4D2 C 2n e 2 4D2 ðC 2mþ1 þ C 2m Þð2n 1Þ!!2n 2mnþ2 2mþ2 ð2n 1Þ!!2 2mnþ2 D D n¼1 n¼1
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
(
m 2m2nþ2 X 1 nx þ mþ1 ð2m þ 1Þ!!2mþ1 þ C 2n 2mþ2 ð2n 1Þ!!2 2mþ2 D D D2mnþ2 n¼1 ( ) 2 X 2n x2mþ2 mþ1 x2m2nþ2 1þ x þ C 2mþ2 ð2n 1Þ!!2n 2mnþ2 : ¼ e 2 4D2 2mþ2 D D n¼1 12þ x
¼e
2
4D2
x2mþ2
123
)
Eq. (3.6) has been proved. That is, Eq. (3.5) is established 2 0
p kþ12 pffiffiffiffi A þ A 2 K A; z; A0 ; z0 ¼ ðAA0 Þ p 8 4
!12
2
@ 2k @zk @Z 0 k
where A; A0 2 M p4 ¼ AjjargAj < p4 ; z; z0 2 C; k 2 Zþ .
Eq. (3.5) implies that K A; z; A0 ; z0
e
12þ
½AþA0 þ2ðZZ0 Þi 4ðA2 þA0 2 Þ ;
h
is analytic on M p4 C as a function in ðA; zÞ, and it is anti-analytic on M p4 C as a
function in ðA0 ; z0 Þ. For the function f 2 L2 ðRÞ, in particular, we find that the image ðT wav f Þða; xÞ can be extended analytically onto the space M p4 C with the form ðT wav f ÞðA; zÞ, and the image ðT wav f Þða; xÞ can be characterised as the members of the Hilbert space Hk admitting the reproducing kernel function (3.3). 4. The reproducing kernel space when the scale factor is fixed Now, we merely consider the wavelet transform for any fixed A 2 M p4 . Then, we obtain the corresponding reproducing 0 kernel by setting A ¼ A 2
pffiffiffiffiffiffiffi ½aþðZZ0 Þi
12þ 1 1 p 2p 2 @ 2k 2 kþ2 2 2 2 0 0 2ða2 b2 Þ 0 K A; z; A ; z ¼ K A z; z ¼ ða þ b Þ ða b Þ e : 8 @zk @Z 0 k
For any fixed A 2 M p4 , and for any f 2 L2 ðRÞ, the image ðT wav f Þða; xÞ of the wavelet transform (3.1) can be characterised by the members of Hilbert space HkA admitting the above reproducing kernel and composed with all entire functions. Theorem 4.1. Let
pffiffiffiffiffiffiffi 2 0 1 1 1 ½aþðZZ Þi p 2p 2 2 kþ 2 þ K F ðkÞ z; z0 ¼ ða þ b Þ 2 ða2 b Þ 2 e 2 2ða2 b2 Þ ; 8
k 2 Zþ :
The function K F ðkÞ z; z0 is the reproducing kernel of Hilbert space HF ðkÞ composing all entire function hðzÞ with finite norms
khðzÞkF ðkÞ
8 > <
91 2 ( ) pffiffiffi > ZZ = a 2 4 2 1 2ðy 2Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy ¼ jhðzÞj exp 2 > > 2 a2 b :p2 pða2 b2 Þða2 þ b2 Þ2kþ1 C ;
and, we have the isometric identity ðkÞ
kh ðzÞkHK ¼ khkHF ðkÞ : A
Proof. Let
sðzÞ ¼
8 91
: 32ða2 b Þ ;
2 þ4azi2z2 4ða2 b2 Þ
14þa
e
and then, we have
pffiffiffiffiffiffiffi 2 0 1 1 1 ½aþðZZ Þi p 2p 2 2 kþ 2 þ K F ðkÞ z; z0 ¼ sðzÞsðz0 ÞK z; z0 ¼ ða þ b Þ 2 ða2 b Þ 2 e 2 2ða2 b2 Þ ; 8 where
K z; z0 ¼
zz0 a2
b
2
:
By Lemma 2.1 and Definition 2.3, we find that the function K F ðkÞ z; z0 is the reproducing kernel of the Hilbert space HF ðkÞ with the finite norm
124
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
( ( ) )12 2 ZZ 2 hðzÞ 1 ¼ hðzÞ exp jzj dxdy khðzÞkF ðkÞ ¼ sðzÞ 2 pða2 b2 Þ C sðzÞ a2 b Hk By the Euler formula
eðaþibÞx ¼ eax ðcosbx þ isinbxÞ and
jsðzÞj2 ¼ jsðzÞ2 j: we can obtain
khðzÞkF ðkÞ
8 > <
91 2 ( pffiffiffi ) > ZZ = a 2 4 2 1 2 y2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy ¼ jhðzÞj2 exp : 2 > > 2 a2 b :p2 pða2 b2 Þða2 þ b2 Þ2kþ1 C ;
Then, we have the isometric identity ðkÞ
kh ðzÞkHK ¼ khkHF ðkÞ :
A
Note that for any fixed A 2 M p4
1 tz ;z2C A2 w A
is complete in L2 ðRÞ. Then, we have the following characterisation of ðT wav f ÞA ðzÞ. Theorem 4.2. For any fixed A 2 M p4 ; k 2 Zþ , the image ðT wav f ÞA ðzÞ of a family of complex wavelet transform for f 2 L2 ðRÞ are characterised with properties that ðT wav f ÞA ðzÞ are entire functions that satisfy the following formula
ZZ
jðT wav f ÞA ðzÞj2 exp ðkÞ
C
( 2 ) 1 2 y 2a dxdy < 1: 2 2 a2 b
Furthermore, then we obtain the isometrical identity
kf k2 ¼
( pffiffiffi 2 ) ZZ 4 2 1 2 y 2a ðkÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy: jðT wav f ÞA ðzÞj2 exp 2 2kþ1 2 C a2 b p2 pða2 b2 Þða2 þ b2 Þ
ð4:1Þ
Proposition 4.1. The images f ðpÞ of the linear transform (2.1) for H form precisely the functional Hilbert space Hk admitting the reproducing kernel kðp; qÞ defined by (2.2), which is uniquely determined by the reproducing kernel kðp; qÞ. Then, we have the inequality
kf kHk 6 kfkH : Furthermore, for any f 2 Hk , there exists a uniquely determined member f of H such that
f ðpÞ ¼ ðf ; hðpÞÞH and
kf kHk ¼ kf kH : We shall assume that the norms in H and Hk are realised in terms of r-finite positive measures dm and dl on measurable spaces T and E in the following manners
H ¼ L2 ðT; dmÞ;
Hk L2 ðE; dlÞ:
Thus, we shall consider the integral transform
f ðpÞ ¼
Z
FðtÞhðt; pÞdmðtÞ;
T
where hðt; pÞ is a complex-valued function on T E with
hð; pÞ 2 L2 ðT; dmÞ
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
125
and for the functions F satisfying
F 2 L2 ðT; dmÞ; the corresponding reproducing kernel is
kðp; qÞ ¼
Z
hðt; qÞhðt; pÞdmðtÞ: T
Proposition 4.2. Suppose that there exists an exhaustion fEN g1 N¼1 of dl-measurable sets of E satisfying (1) E1 E2 , (2) [1 E ¼ E, R N¼1 N (3) EN Kðp; qÞdlðpÞ < 1. For a member f of Hk , if
Z
f ðpÞhðt; pÞdlðpÞ 2 L2 ðT; dmÞ;
EN
we have the sequence
Z
f ðpÞhðt; pÞdlðpÞ
1
EN
m; N¼1
that converges strongly to F in Proposition 4.1. That is
F ðtÞ ¼ s lim
Z
N!1
f ðpÞhðt; pÞdlðpÞ
EN
for the function F satisfying
f ðpÞ ¼
Z
F ðtÞhðt; pÞdlðtÞ
T
and
kf kHk ¼ kF kL2ðT; dlÞ : Next, we shall derive a characterisation of the images ðT wav f ÞA ðxÞ as real variable functions. By the Plancherel theorem, the isometrical identity (4.3) and Proposition 4.1, we have Theorem 4.3. For any fixed A 2 M p4 ; k 2 Zþ , and for f 2 L2 ðRÞ, the image ðT wav f ÞA ðxÞ of (3.1) are characterised by the properties 1 that are of class C ðRÞ. Then, we furthermore obtain the isometrical identity
kf k2 ¼
2 pffiffiffi 2 n Z nk wav 1 X 4 2 2n ða2 b Þ @ ðT f ÞðxÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx; nk 2kþ1 @x n! R n¼0 p2 pða2 þ b2 Þ
ð4:4Þ
where
@ 1 ðT wav f ÞðxÞ ¼ @x1
Z
x
ðT wav f ÞðlÞdl
infty
and
! @ 2 @ 1 @ 1 ¼ : @x2 @x1 @x1 From the identities (4.3) and (4.4), we obtain the family of the complex wavelet transform inversion formulas for (3.1) by Proposition 4.1, in a general argument. Theorem 4.4. In the complex wavelet transform (3.1), for any fixed A 2 M p4 ; k 2 Zþ , we have the family of the complex wavelet transform inversion formulas
pffiffiffiffiffiffi1 1 ðkÞ 2ðyaÞ2 ZZ 2 4 2e2 A2 tz ðkÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT wav f Þ ðzÞw e a2 b2 dxdy: 2kþ1 N!1 A EN p2 pða2 b2 Þða2 þ b2 Þ
f ¼ lim
ð4:5Þ
126
S. Li et al. / Applied Mathematics and Computation 233 (2014) 116–126
For any compact exhaustion fEN g1 N¼1 of C, and
pffiffiffiffiffiffi1 1 ! 2 n Z N X 4 2e2 A2 2n ða2 b Þ @ nk ðT wav f ÞðxÞ @ nk tx dx; f ¼ lim qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2kþ1 N!1 @xnk n! @xnk A R n¼0 p2 pða2 þ b2 Þ
ð4:6Þ
in the sense of strong convergence in L2 ðRÞ. 5. Conclusions In this paper, an admitting complex wavelet is constructed using a convolution formula, and the image space of the complex wavelet transform is characterised by the reproducing kernel throry. The results of Ref.[18] have been popularised and supported. Accordingly, this article should serve as the basis for additional research and further useful results. Acknowledgments The work is supported by a Grant from the Youth Natural Science Foundation of Heilongjiang Province, No. QC2011C103. References [1] X.W. Zhang, J.F. He, X.B. Zhang, The parameter estimation algorithm research for weak sinusoidal signal, J. Numer. Methods Comput. Appl. 3 (2012) 222–229. [2] L.Q. Li, Y.Y. Tang, Wavelet–Houch transform with applications in edge and target detections, Int. J. Wavelets Multiresolution Inf. Process. 4 (1992) 567– 587. [3] ZH.X. Cheng, SH.ZH. Yang, X.X. Feng, Wavelet Analysis Theory. The Algorithm, Progress and Application, The Defense Industry Press, Beijing, China, 2007. 05: 145–150. [4] F.Z. Geng, A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method, Appl. Math. Comput. 218 (8) (2011) 4211–4215. [5] N. Hauyashi, S. Saitoh, Analyticity and smoothing effect for the Schrodinger equation, Ann. Inst. Henripoincare 52 (1990) 163–173. [6] B.Y. Wu, X.Y. Li, Application of reproducing kernel method to third order three-point boundary value problems, Appl. Math. Comput. 217 (7) (2010) 3425–3428. [7] H. Long, X.J. Zhang, Construction and calculation of reproducing kernel determined by various linear differential operators, Appl. Math. Comput. 215 (2) (2009) 759–766. [8] M. Mouattamid, Recursive producing kernels Hilbert space using the theory of power kernels, Anal. Theory Appl. 2 (2012) 111–124. [9] J.S. Chen, C. Pan, H.P. Wang, A lagrangian reproducing kernel particle method for metal forming analysis, Comput. Mech. 6 (1998) 289–307. [10] N.R. Aluru, A reproducing kernel particle method for meshless analysis of microelectro mechanical systems, Comput. Mech. 23 (1999) 324–338. [11] N.R. Aluru, A point collocation method based on reproducing kernel approximations, Int. J. Numer. Methods Eng. 47 (2000) 1083–1121. [12] S. Saitoh, Reproducing kernels of the direct product of two Hilbert space, Riazi. J. Kar. Math. Assoc. 4 (1982) 1–20. [13] S. Saitoh, Representations of the norms in Bergman–Selberg spaces on strips and half planes, Complex Variables 19 (1992) 231–241. [14] S. Saitoh, Quadratic norm inequalities deduced from the theory of reproducing kernels, Linear Algebra Appl. 93 (5) (1987) 171–178. [15] L.J. Gu, C.X. Deng, Q.Y. Ling, Expressing wavelet transform with reproducing kernel, J. Math. 28 (2008) 507–513. [16] C.X. Deng, Q.Y. Ling, L.J. Gu, Characterization of image space of a wavelet transform, Int. J. Wavelets Multiresolution Inf. Process. 4 (2006) 547–557. [17] Q.Y. Ling, C.X. Deng, J. Hou, Characterization of image space of Gauss wavelet transform, ACTA Math. Sinica Chin. Ser. 51 (2008) 225–234. [18] S.S. Li, C.X. Deng, Z.X. Fu, The isometric and the inversion formulas of image space of Cgau wavelet transform, Chin. Ann. Math. 32 (2011) 121–128. [19] L.P. Castro, E.M. Rojas, S. Saitoh, Inversion from different kinds of information and real inversion formulas of the Laplace transform from a finite number of data, Math. Eng. Sci. Aerosp. 02 (2010) 181–190. [20] L.P. Castro, H. Fujiwara, S. Saitoh, Y. Sawano, A. Yamada, M. Yamada, Fundamental error estimates inequalities for the Tikhonov regularization using reproducing kernels, Int. Ser. Numer. Math. Inequalities Appl. 02 (2012) 87–101. [21] I. Daubechies, Ten lectures on wavelets, Soc. Ind. Appl. Math. 1 (1992) 256–263.