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1 引言小波分析是结合泛函分析、应用数学、逼近论、调和分析、广义函数论等数学知识的结晶,具有深刻的理论意义和广泛的应用范围,被称为"数学显微镜".基于其多分辨分析的特点以及在时、频两域都具有表征信号局部特征的功能,应用它可以解决许多Fourier变换不能解决的难题,为工程应用提供了一种新的、更有效的分析工具[1]. 相似文献
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1引言1950年N.Aronszjan发表的一篇综述性文章《Theory of reproducing kernels》标志再生核理论的初步形成.由于再生核有许多良好的计算性质,S.Saitoh总结并深入研究再生核基本理论,进一步拓展了再生核的应用领域;徐利治把再生核应用于L~2(B)(B是复平面上的一个区域)中解析函数重积分降维问题,并提出了一个能对一些预先给出的 相似文献
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《Journal of Computational and Applied Mathematics》2006,197(1):218-232
We examine the use of wavelet packets for the fast solution of integral equations with a highly oscillatory kernel. The redundancy of the wavelet packet transform allows the selection of a basis tailored to the problem at hand. It is shown that a well chosen wavelet packet basis is better suited to compress the discretized system than wavelets. The complexity of the matrix–vector product in an iterative solution method is then substantially reduced. A two-dimensional wavelet packet transform is derived and compared with a number of one-dimensional transforms that were presented earlier in literature. By means of some numerical experiments we illustrate the improved efficiency of the two-dimensional approach. 相似文献
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在以再生核Hilbert空间为连续小波变换像空间的基础上,针对Cgau小波(复数形式的Gauss小波),给出了其小波变换像空间的再生核的具体表达式.当固定尺度因子时,利用再生核空间理论,对Cgau小波变换像空间做了具体描述,分别给出了Cgau小波变换像空间中的等距变换和反演公式,这为进一步研究一般的小波变换像空间提供了理论基础. 相似文献
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《Applied and Computational Harmonic Analysis》2014,36(1):158-166
Gabor functions, Gaussian wave packets, are optimally localized in time and frequency, and thus in principle ideal as (frame) basis functions for a wavelet, windowed Fourier or wavelet-packet transform for the detection of events in noisy signals or for data compression. A major obstacle for their use is that a tailored efficient operator acting on the transform coefficients for altering the width of the wave packets does not exist. However, by virtue of a curious property of the Gabor functions it is possible to change the width of the wave packets using just one-dimensional convolutions with very short kernels. The cost of a wavelet-type transform based on the scheme presented below is similar to that of a low order wavelet transform for a compact kernel and significantly less than the algorithme à trous. The scheme can hence easily be employed for the processing of signals in real time. 相似文献
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This paper is concerned with reconstruction problems arising in the context of radar signal analysis. The goal in radar is to obtain information about objects by emitting certain signals and analyzing the reflected echoes. In this paper, we shall focus on the general wideband model for radar echoes and on the case of continuously distributed objects D (reflectivity density). In this case, the echo is given by an inverse wavelet transform of the density D where the role of the analyzing wavelet is played by the transmitted signal. However, the null space of an inverse wavelet transform is nontrivial, it is described by the corresponding reproducing kernel. Following the approach of Naparst [14] and Rebolla-Neira et al. [16], we suggest to treat this problem by transmitting not just one signal but a family of signals. Indeed, a reconstruction formula for one- and 2-dimensional reflectivity densities can be derived, provided that the set of outgoing signals forms an orthogonal basis or – more general – a frame. We also present some rigorous error estimates for these reconstruction formulas. The theoretical results are confirmed by some numerical examples. 相似文献
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This article presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The
essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by
locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotationinvariant
pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform
similar to the algorithms known from classical tensor product wavelet theory. 相似文献
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Michael Schreiner 《PAMM》2007,7(1):1050401-1050402
This paper presents a method for approximating spherical functions from discrete data of a block–wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotation–invariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from classical tensor product wavelet theory. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Hari M. Srivastava Komal Khatterwani S. K. Upadhyay 《Mathematical Methods in the Applied Sciences》2019,42(9):3103-3122
In the present paper, a fractional wavelet transform of real order α is introduced, and various useful properties and results are derived for it. These include (for example) Perseval's formula and inversion formula for the fractional wavelet transform. Multiresolution analysis and orthonormal fractional wavelets associated with the fractional wavelet transform are studied systematically. Fractional Fourier transforms of the Mexican hat wavelet for different values of the order α are compared with the classical Fourier transform graphically, and various remarkable observations are presented. A comparative study of the various results, which we have presented in this paper, is also represented graphically. 相似文献
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1.IntroductionInthispaper,westudytheFredholmintegro-differentialequationbythewaveletmethod.Theapplicationsoftheequationinimagerestorationcouldbefoundin[101.ForthehistoryofnumericalmethodsfortheFredholmintegro-differentialequations,wereferto[4].FOllow... 相似文献
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Carnovale 《Constructive Approximation》2008,18(3):309-341
The investigation of a q -analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, with special attention to
the approximation of functions by means of the convolution. A new space of functions that forms an increasing chain of algebras
(with respect to the q -convolution), depending on a parameter s>0 , is constructed. For a special value of the parameter the corresponding algebra is commutative and unital, and is shown
to be the quotient of an algebra studied in a previous paper modulo the kernel of a q -analogue of the Fourier transform. This result has an analytic interpretation in terms of analytic functions, whose q -moments have a (fast) decreasing behavior and allows the extension of Koornwinder's inversion formula for the q -Fourier transform. A few results on the invertibility of functions with respect to the q -convolution are also obtained and they are applied to the solution of certain simple linear q -difference equations with polynomial coefficients. 相似文献
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Clifford-hermite wavelets in euclidean space 总被引:1,自引:0,他引:1
Specific kernel functions for the continuous wavelet transform in higher dimension and new continuous wavelet transforms are
presented within the framework of Clifford analysis. Their relationship with the heat equation and the newly introduced wavelet
differential equation is established. 相似文献
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《Applied and Computational Harmonic Analysis》2020,48(3):811-867
In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense. 相似文献
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R. S. Pathak 《Applicable analysis》2013,92(7):1324-1332
In the present paper, we discuss about extension of the wavelet transform on distribution space of compact support and develop the Paley–Wiener–Schwartz type theorem for the wavelet transform on the same. Furthermore, Paley–Wiener–Schwartz type theorem for the wavelet transform is also established using the relation between the wavelet transform and double Fourier transform. 相似文献
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Türker Nazmi Erbil 《Mathematical and Computer Modelling of Dynamical Systems: Methods, Tools and Applications in Engineering and Related Sciences》2013,19(6):493-513
In this paper, we utilize wavelet transform to obtain dynamical models describing the behaviour of fluid flow in a local spatial region of interest. First, snapshots of the flow are obtained from experiments or from computational fluid dynamics (CFD) simulations of the governing equations. A wavelet family and decomposition level is selected by assessing the reconstruction success under the resulting inverse transform. The flow is then expanded onto a set of basis vectors that are constructed from the wavelet function. The wavelet coefficients associated with the basis vectors capture the time variation of the flow within the spatial region covered by the support of the basis vectors. A dynamical model is established for these coefficients by using subspace identification methods. The approach developed is applied to a sample flow configuration on a square domain where the input affects the system through the boundary conditions. It is observed that there is good agreement between CFD simulation results and the predictions of the dynamical model. A controller is designed based on the dynamical model and is seen to be successful in regulating the velocity of a given point within the region of interest. 相似文献
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Hong-xia Wang Li-zhi Cheng 《应用数学学报(英文版)》2005,21(3):459-468
Fast wavelet transform algorithms for Toeplitz matrices are proposed in this paper. Distinctive from the well known discrete trigonometric transforms, such as the discrete cosine transform (DCT) and the discrete Fourier transform (DFT) for Toeplitz matrices, the new algorithms are achieved by compactly supported wavelet that preserve the character of a Toeplitz matrix after transform, which is quite useful in many applications involving a Toeplitz matrix. Results of numerical experiments show that the proposed method has good compression performance similar to using wavelet in the digital image coding. Since the proposed algorithms turn a dense Toeplitz matrix into a band-limited form, the arithmetic operations required by the new algorithms are O(N) that are reduced greatly compared with O(N log N) by the classical trigonometric transforms. 相似文献