Multiresolution approximations and wavelet orthonormal bases of L^2(R)
TLDR
In this paper, the authors study the properties of multiresolution approximation and prove that it is characterized by a 2π periodic function, which is further described in terms of wavelet orthonormal bases.Abstract:
A multiresolution approximation is a sequence of embedded vector spaces V j jmember Z for approximating L 2 (R) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 2π periodic function which is further described. From any multiresolution approximation, we can derive a function ψ(x) called a wavelet such that √ 2 j ψ(2 j x −k) (k ,j)member Z 2 is an orthonormal basis of L 2 (R). This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space H s .read more
Citations
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Intermittency, local isotropy, and non‐Gaussian statistics in atmospheric surface layer turbulence
TL;DR: In this article, a relation between the &h-order structure function and wavelet coefficients is derived to examine deviations from the classical Kolmogorov theory for velocity and temperature in the inertial subrange.
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Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for High Order Methods
Björn Sjögreen,Helen C. Yee +1 more
TL;DR: To minimize the tuning of parameters and physical problem dependence, new sensors with improved detection properties are proposed, derived from utilizing appropriate non-orthogonal wavelet basis functions and they can be used to completely switch off the extra numerical dissipation outside shock layers.
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Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N
TL;DR: In this article, the wavelets originating from multiresolution analysis of scaleN give rise to certain representations of the Cuntz algebras, and conversely the wavelet can be recovered from these representations.
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Principal‐component analysis of multiscale data for process monitoring and fault diagnosis
Seongkyu Yoon,John F. MacGregor +1 more
TL;DR: In this article, a multivariate statistical process control (MSPC) approach is presented for process monitoring and fault diagnosis based on principal component analysis (PCA) models of multiscale data.
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Wavelets and pre-wavelets in low dimensions
TL;DR: Chui et al. as mentioned in this paper consider multiresolutions generated by suitable compactly supported and symmetric functions ϑ and explicitly construct 2s − 1 compactly-supported functions such that the translates ϑμ(· − j), j∈ Z s, μ ϵ Z 2s⧹ 0, are an unconditional basis for W0.
References
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A theory for multiresolution signal decomposition: the wavelet representation
TL;DR: In this paper, it is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2 /sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.
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Orthonormal bases of compactly supported wavelets
TL;DR: This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.
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Decomposition of Hardy functions into square integrable wavelets of constant shape
A. Grossmann,J. Morlet +1 more
TL;DR: In this article, the authors studied square integrable coefficients of an irreducible representation of the non-unimodular $ax + b$-group and obtained explicit expressions in the case of a particular analyzing family that plays a role analogous to coherent states (Gabor wavelets) in the usual $L_2 $ -theory.
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Exact reconstruction techniques for tree-structured subband coders
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Analysis of sound patterns through wavelet transforms
TL;DR: The main features of so-called wavelet transforms are illustrated through simple mathematical examples and the first applications of the method to the recognition and visualisation of characteristic features of speech and of musical sounds are presented.