The Isometric Identities And The Inversion Formulas Of Image Space Of Complex Continuous Wavelet Transform

Wavelet analysis develops quickly in current mathematical filed. It is not only a bril-liant combination of pure mathematics and applied mathematics after Fourier transform which has been called mathematics microscope. It is also one of the most obvious achieve-ments of harmonic analysis in the past half century. Wavelet analysis can pick up useful information from signal effectively, and it also can solve lots of difficult subjects that can’t be solved by Fourier transform. Now, it has made most important valuable achievement in the filed of signal analysis, image process, sound compound, solution of equations and so on. We all know that there is redundancy of information in wavelet transform, wavelet transform coefficient has certain relevance in wavelet transform image plane. We can also find that reproducing kernel Hilbert space is the basic of wavelet transform, and relevance region’s magnitude is given by reproducing kernel. What’s more, we can prove that its magnitude will decrease if the scale decreases. Therefore, we can conclude that reproduc-ing kernel space plays an important role in the reconstruction of the continuous wavelet transform.Daubechies.I pointed out in her book’Ten Lectures on Wavelets’that the image space of continuous wavelet in L2(R) is a closed subspaceL2(R×R; a-2dadx), its a reproducing kernel Hilbert space. And the element of the reproducing kernel Hilbert space can be denoted by the reproducing kernel of the space.In particular, the continuous wavelet basis function whose scale and translation are continuous changes is a complete base,we can use the reproducing kernel to describe the correlation of two basis function.By the reconstruction of the kernel,continuous wavelet is a base which has high redundancy. So that, There is redundancy of information in the process of wavelet transform. There is correlation between the wavelet coefficient and the reproducing kernel is the basis of wavelet transform. The reproducing kernel measure the choice of each wavelet space and scale, so it will be helpful to select the most suitable wavelet for a given problem of wavelet.At the same time, the wavelet basis func-tion is relevant, that means that there is correlation in the wavelet coefficient Twav f(a,x) and Twav f(a’,x’) in the different points (a,x) and (a’ x’),this This agree with that the image space of wavelet is a reproducing kernel space.In conclusion, for any random signal, the size of the related area of the continuous wavelet transform can be given by reproducing kernel, namelywhere, is the wavelet admissibility condition.And with the decrease of the scale, the areas of continuous wavelet transform also decrease. Therefore, for the different wavelet base, the image space of different wavelet transform is determined by the reproducing kernel func-tion. So we can see that reproducing kernel function is a powerful tool to determine the properties of wavelet basis function.For example, the effective measure of reproducing ker-nel can measure the influence for base function which depending on four parameters of the wavelet transform, and it can determine the minimum range related four parameter.When the wavelet reconstruction or discretization processing,we shall require the sampling points less as far as possible, and keep as much as useful information of signals. Therefore, we can discuss the properties of wavelet transform from the properties of reproducing kernel function. So that,Without reproducing kernel function, the continuous wavelet transform cannot be rebuilt, not to mention the wavelet analysis method.There are many kinds of wavelet basis in Wavelet analysis, The emergence of so many wavelet sometimes makes users don’t know how to choice.In practice, how to choose a good wavelet base will be a most pressing problem. We discuss wavelet transform purely before, and don’t consider the image space of wavelet transform. Due to the image space of wavelet transform is a reproducing kernel space,and the elements of Reproducing kernel space can be represented by a reproducing kernel of the space. So that,by the reproducing kernel function of image space of wavelet transform, further understanding the structure of the continuous wavelet transform as the space is especially important.This is a good try to discuss the theory of wavelet analysis from the theory of reproducing kernel.This thesis mainly studies the image space of the complex Gauss wavelet transform,and according to the nature of the reproducing kernel,we give the isometric transformation and inverse formula of the image space respectively. The thesis is divided into four parts. The basis part of the paper is the first and second chapter. In this part, we give the overview of the wavelet analysis and the theory of reproducing kernel, at the same time we gave the research background, significance and research content in this paper. At last we use reproducing kernel to discuss the image space of wavelet transform as a brief review.And The fifth chapter of the paper is the conclusion and prospect.The core part is the third and the fourth chapter of the thesis, and this paper we give preliminary knowledge of reproducing kernel and wavelet transform.I.The isometric transformation, inversion formula,and sampling theorem of the image space of complex Gauss waveletThis part mainly to discuss the image space of complex Gauss wavelet transformation, we obtained the reproducing kernel of the wavelet transform, and we use the specific nature of reproducing kernel to get the isometric transformation, inversion formula,and sampling theorem.The main contents are as follows:1. The simply described of the image space of Wavelet transformLetφ(t) is a square integrable function,then satisfiedφ(t)∈L2(M), Defines the Fourier transform isItΦ(ω) satisfies the admissibility conditionsthen we called Φ(t) is Mother wavelet function or basic wavelet function.That required Consider about wavelet function departmentΦa,x(t) which after expansion and trans-lation of the basic wavelet function psi(t)then we find norm Φa,x(t)and Φ(t) equality inL2(R),then we haveThe continuous wavelet transform of wavelet function is given byFor the continuous wavelet transform formula above, there is inner product theorem is as follows:ThenTherefore, the reproducing kernel function of continuous wavelet transform can be ex-pressed by the type:By reproducing kernel theory, we know,a positive definite matrix K(a,x;a’,x’)can only determine a reproducing kernel spaceHk which has the reproducing kernel K(a,x;a’,x’). And the Hilbert space Hk is the image space of continuous wavelet transform above.This space is proved to be a subspace of L2(R).And we are interested in this space.2. Cgau wavelet transform and the image space of Cgau wavelet transform Complex Gauss mother wavelet function is its Fourier transform isthen we easy to know Φ(x) Satisfits admissibility conditions.Real part of Cgau wavelet is shown in Figure1; Imaginary part of Cgau wavelet is shown in Figure2.Cgau continuous wavelet function isFor(?)f∈L2(R),Cgau wavelet transform is3.Reproducing kernel function of image space of Cgau wavelet transformIn order to get reproducing kernel function of image space of Cgau wavelet transform,we use the redundancy of wavelet transform,at the same time to consider the symmetry aboutaWhena>0,we extend the type (7) forwhere f∈L2(R),and we mark the image space of wavelet transform (8)by H, whereA=a+bi, Z=x+yi,a,b,x,y∈R,a> O.The reproducing kernel of the image space H of wavelet transform (8)iswhereA’=a’+b’i, Z’=x’+y’i,a’,b’,x’,y’∈R,a’> O.And we can obtain Theorem3.4The reproducing kernel function of image space of wavelet transform (8)is K(A,Z;A’,Z’) whereBy the expression of reproducing kernelK(A, Z; A’, Z’),we find K(A, z; A’,z’) is analytic on△(π/4)×C as a function in (A,z), and it is anti-analytic on△(π/4)×C as a function in (A’,z’). For the function f∈L2(R), in particular, we find that the im-age (Twavf)(a,x) can be extended analytically onto the space△(π/4)×C with the form (Twav f)(A,z), and the image (Twavf)(a, x) can be characterised as the members of the Hilbert space Hk admitting the reproducing kernel function by thm2.Furthermore, for{A-1/2Φ(t-Z/A);A∈△(π/4),z∈G is complete in H,so wc have the isometric identityby (),Further identity can be obtainedAlthough its expression is not simple, the norm of the wavelet transform ()which defined on△(π/4)×Ccan still give by (Twav f)(a, x) on R+×R. Therefore (11) is a very meaningful identity.4.The description of the image space of Cgau When the Scale Factor is FixedBy the nature of the reproducing kernel,for the reproducing kernel Hilbert space H^on E and for any nonzero complex functions (p) onEis a reproducing kernel of Hilbert space.Hks, where Hkis constituted type as function fs(p) on E belowand Hks has the following inner productAt the same time, for the Hilbert space (?)(λ)(λ>0) comprising all entire functions f(z) with finite norms the function eλCz on C x C is the reproducing kernel.And for any element f(z)∈(?)(λ)andz∈C,u∈Chave f(u)=<f(z),eλuz,this space is called Fock space.In section3.2, We use the properties of Fock reproducing kernel and Fock space, to get finite norm, isometric transformation of image space of Cgau wavelet transform Theorem3.5LetThen Kk(λ) is the reproducing kernel of Hilbert space HD(λ) which is made up all the entire function h(z) and has the finite norm (12)And the image space of Cgau wavelet transform HkA has finite normTheorem3.6For any fixed A∈△(π/4),the image of Cgau wavelet transform (Twav f)(A, Z),for f∈L2(R), satisfyFurther we get the isometric identityInversion formula of Wavelet transform is a difficult point, we introduce the heat conduction equationThe solution iswhere f∈L2(R). And we sent imageu(x,t) analytic continuation on C, using the existing isometric identity,spread to the general situation, we get Theorem3.9For any fixed A∈△(π/4) in wavelet transform (), for any tight exhaustive sequence{EN}> N=1,2...∞onC,we can get inversion formulaII. The Isometric Identities and Inversion Formulas of Complex Contin-uous Wavelet TransformsThis part promote with the third chapter,then we obtain the isometric identities and inversion formulas of complex continuous wavelet transforms.The main contents are as follows:1. The simply description of the space of Wavelet transform Theorem4.2Let Φ(t) be a basic wavelet, g(t)∈L1(R) be a bounded function, and then its convolution Φ*g(t) is also a basic wavelet. Proposition4.3Let f(t)=e-it-2t2, and g(t)=f(k-1)(t), where k∈Z+is a positive integer, and Φ1(t)=e-it-2t2(-i-4t), then the convolution function Φ(t)=Φ1*g(t) is a basic wavelet.The Fourier transform of Φ(t) isThen,we can find the conclusion by the inversion formula of Fourier transform. When k=1,Φ(t) below is the cgau wavelet (complex Gauss wavelet)functionWhen k=2,Φ(t) below is the complex Mexihat wavelet functionBy the symmetry in a, we merely consider the wavelet transform for a>0. We shall consider the wavelet transform in the complex form beloworwhere A=a+bi, z=x+yi, a, b, x, y∈R,a>0,and Φ(t) is the wavelet function in Proposition4.3. Proposition4.4The reproducing kernel function for the image space of the wavelet transform(8)is given by (18)where,K∈Z+where D≠0,D∈C.Theorem4.6The reproducing kernel function of the image space wavelet transform(8) can be expressed further in the following form K(A,z;A’,z’]K(A,z;A’,z’)=At this time,A,A’∈△(π/4)={AllargAl<π/4},z,z’∈C,K∈Z+The equation(20)implies that K(A,z;A’,z’)is analytic on△(π/4)×C as a function in(A,z),and it is anti-analytic on△(π/4)×C as a function in(A’,z,).For the function f∈L2(R),in particular,we find that the image(Twav f)(a,z)can be extended analytically onto the space△(π/4)×C with the form(Twav f)(A,z),and the image(Twav f)(a,x)can be characterised as the members of the Hilbert space Hk admitting the reproducing kernel function (18).2. The Reproducing Kernel Space When the Scale Factor is FixedNow,we merely consider the wavelet transform for any fixed A∈△(π/4).Then,we obtain the corresponding reproducing kernel by setting A=A (21)When fixed A∈△(π/4), and for any f∈L2(R), the image (Twav f)(A, Z)=(Twav f)A(Z) of the wavelet transform (8)can be characterised by the members of the Hilbert space HKA admitting the above reproducing kernel and composed with all entire functions.Further description of the space, we get the following theoremTheorem4.7LetThe function KF(λ)(z, z’) is the reproducing kernel of Hilbert space HF(λ) composing all entire function h(z) with finite normsand,we have the isometric identityTheorem4.8For any fixed A∈△A(π/4), k∈Z+, the image (Twav f)A(z) of a family of com-plex wavelet transform for f∈L2(R) are characterised with properties that (Twav f)A(z) are entire functions that satisfy the following formulaFurthermore, then we obtain the isometrical identity Theorem4.11For any fixed A∈△(π/4), k∈Z+, and for f∈L2(R), the image (Twav f)A(x) of (8) are characterised by the properties that are of class C∞(R). Then,we furthermore obtain the isometrical identitywhereandTheorem4.12In the complex wavelet transform (8), for any fixed A∈△(π/1), k∈Z+we have the family of the complex wavelet transform inversion formulasFor any compact exhaustion{EN}∞N=1of C, andin the sense of strong convergence in L2(R).