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Shannon Wavelets Theory Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2008, Article ID 164808, 24 pages doi:10.1155/2008/164808 Research Article Shannon Wavelets Theory Carlo Cattani Department of Pharmaceutical Sciences (DiFarma), University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy Correspondence should be addressed to Carlo Cattani, [email protected] Received 30 May 2008; Accepted 13 June 2008 Recommended by Cristian Toma Shannon wavelets are studied together with their differential properties known as connection coefficients. It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization R ff coincides with the Shannon wavelet reconstruction of L2 functions. The di erential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon ∞ wavelets are C -functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the C -functions. Copyright q 2008 Carlo Cattani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Wavelets 1 are localized functions which are a very useful tool in many different applications: signal analysis, data compression, operator analysis, and PDE solving see, e.g., 2 and references therein. The main feature of wavelets is their natural splitting of objects into different scale components 1, 3 according to the multiscale resolution analysis. R For the L2 functions, that is, functions with decay to infinity, wavelets give the best approximation. When the function is localized in space, that is, the bottom length of the function is within a short interval function with a compact support, such as pulses, any other reconstruction, but wavelets, leads towards undesirable problems such as the Gibbs phenomenon when the approximation is made in the Fourier basis. In this paper, it is shown that Shannon wavelets are the most expedient basis for the analysis of impulse functions pulses4. The approximation can be simply performed and the reconstruction by Shannon wavelets range in multifrequency bands. Comparing with the Shannon sampling theorem where the frequency band is only one, the reconstruction by Shannon wavelets can be done for functions ranging in different frequency bands. Shannon sampling theorem 5 plays a fundamental role in signal analysis and, in particular, for the reconstruction of a signal from a digital sampling. Under suitable hypotheses on a given signal function a few sets of values 2 Mathematical Problems in Engineering samples and a preliminary chosen basis made by the sinc function enable us to completely reconstruct the continuous signal. This reconstruction is alike the reconstruction of a function as a series expansion such as polynomial, i.e., Taylor series, or trigonometric functions, i.e., Fourier series, but for the first time the reconstruction in the sampling theorem makes use of the sinc function, that is a localized function with decay to zero. Together with the Shannon sampling theorem and reconstruction, also the wavelets series become very popular, as well as the bases with compact support. It has been recognized that on the sinc functions one can settle the family of Shannon wavelets. The main properties of these wavelets will be shown and discussed. Moreover, the connection coefficients 6–9also called refinable integrals will be computed by giving some finite formulas for any order derivatives see also some preliminary results in 2, 10–12. These coefficients enable us to define any order derivatives of the Shannon scaling and wavelet basis and it is shown that also the derivatives are orthogonal. 2. Shannon Wavelets Sinc function or Shannon scaling function is the starting point for the definition of the Shannon wavelet family 11. It can be shown that the Shannon wavelets coincide with the real part of the harmonic wavelets 2, 10, 13, 14, which are the band-limited complex functions 4πi2nx−k 2πi2nx−k def e − e ψnx ≡ 2n/2 , 2.1 k 2πi2nx − k with n, k ∈ Z. Harmonic wavelets form an orthonormal basis and give rise to a multiresolution analysis 1–3, 14, 15. In the frequency domain, they are very well localized and defined on compact support intervals, but they have a very slow decay in the space variable. However, in dealing with real problems it is more expedient to make use of real basis. By focussing on the real part of the harmonic family, we can take advantage of the basic properties of harmonic wavelets together with a more direct physical interpretation of the basis. Let us take, as scaling function ϕx, the sinc function Figure 1 πix −πix def sin πx e − e ϕxsinc x 2.2 πx 2πix and for the dilated and translated instances sin π2nx − k ϕnx2n/2ϕ2nx − k2n/2 k π2nx − k 2.3 n − − n − eπi 2 x k − e πi 2 x k 2n/2 . 2πi2nx − k The parameters n, k give, respectively, a compression dilation of the basic function 2.2 and a translation along the x-axis. The family of translated instances {ϕx − k} is an orthonormal basis for the banded frequency functions 5Shannon theorem. For this reason, they can be used to define the Shannon multiresolution analysis as follows. The scaling functions do not represent a basis, in a functional space, therefore we need to define a family of Mathematical Problems in Engineering 3 1 ϕx ψx x −π − π π π 2 2 −1 Figure 1: Shannon scaling function ϕxthick line and wavelet dashed line ψx. functions based on scaling which are a basis; they are called the wavelet functions and the corresponding analysis the multiresolution analysis. Let ∞ def 1 − fωfx fxe iωxdx 2.4 2π −∞ ∈ R be the Fourier transform of the function f x L2 and ∞ fx2π fωeiωxdω 2.5 −∞ its inverse transform. The Fourier transform of 2.2 gives us ⎧ ⎨ 1 1 , −π ≤ ω<π, ϕω χω 3π 2π 2.6 2π ⎩ 0, elsewhere, with ⎧ ⎨ 1, 2π ≤ ω<4π, χ ω ⎩ 2.7 0, elsewhere . Analogously for the dilated and translated instances of scaling function it is −n/2 2 − n ω ϕnω e iω k 1 /2 χ 3π . 2.8 k 2π 2n From the given scaling function, it is possible to define the corresponding wavelet function 1, 15 according to the following. 4 Mathematical Problems in Engineering Theorem 2.1. The Shannon wavelet, in the Fourier domain, is 1 − ψω e iωχ2ω χ−2ω. 2.9 2π Proof. It can be easily shown that the scaling function 2.6 fulfills the condition ω ω ϕωH ϕ , 2.10 2 2 which characterizes the multiresolution analysis 1 with ω H χω 3π. 2.11 2 Thus the corresponding wavelet function can be defined as 1, 15 − ω ω ψωe iωH ± 2π ϕ . 2.12 2 2 With Hω/2 − 2π we have − ω ω ψωe iωH − 2π ϕ 2 2 − 1 ω e iωχω 3π − 2π χ 3π 2π 2 2.13 1 − ω e iωχω πχ 3π 2π 2 1 − e iωχ2ω, 2π then analogously with Hω/2 2π we obtain 1 − ψω e iωχ−2ω, 2.14 2π so that 2.9 follows. For the whole family of dilated-translated instances, it is −n/2 2 − n ω −ω ψnω e iω k 1 /2 χ χ . 2.15 k 2π 2n−1 2n−1 The Shannon wavelet function in the real domain can be obtained from 2.9 by the inverse Fourier transform Figure 1 sin πx − 1/2 − sin 2πx − 1/2 ψx πx − 1/2 2.16 − e 2iπx−i eiπx e3iπx ie4iπx , π − 2πx and by the space shift and compression we have the whole family of dilated and translated instances: sin π2nx − k − 1/2 − sin 2π2nx − k − 1/2 ψnx2n/2 . 2.17 k π2nx − k − 1/2 Mathematical Problems in Engineering 5 By summarizing 2.3 and 2.17, the Shannon wavelet theory is based on the following functions 11: sin π2nx − k ϕnx2n/2 , k π2nx − k 2.18 sin π2nx − k − 1/2 − sin 2π2nx − k − 1/2 ψnx2n/2 k π2nx − k − 1/2 in the space domain, and collecting 2.8 and 2.15, we have in the frequency domain −n/2 2 − n ω ϕnω e iωk/2 χ 3π , k 2π 2n 2.19 −n/2 2 − n ω −ω ψnω− e iω k 1/2 /2 χ χ . k 2π 2n−1 2n−1 The inner product is defined as ∞ def f, g ≡ fxgxdx, 2.20 −∞ which, according to the Parseval equality, can be expressed also as ∞ ∞ def f, g ≡ fxgxdx 2π fωgωdω 2πf,g, 2.21 −∞ −∞ where the bar stands for the complex conjugate. With respect to the inner product 2.20, we can show the following theorem 11. Theorem 2.2. Shannon wavelets are orthonormal functions in the sense that n m nm ψk x ,ψh x δ δhk, 2.22 nm with δ ,δhk being the Kroenecker symbols. Proof. n m ψk x ,ψh x n m 2π ψk ω , ψh ω ∞ −n/2 − −m/2 − 2 −iωk 1/2/2n ω ω 2 iωh 1/2/2m ω ω 2π e χ − χ − e χ − χ − dω −∞ 2π 2n 1 2n 1 2π 2m 1 2m 1 −n m/2 ∞ − − 2 −iωk 1/2/2n iωh 1/2/2m ω ω ω ω e χ − χ − χ − χ − dω 2π −∞ 2n 1 2n 1 2m 1 2m 1 2.23 which is zero for n / m.
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