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Ternary wavelets and their applications to signal compression

100%
Mustafa G., Chen F., Huang Z.
International Journal of Applied Mathematics and Computer Science
|
2004
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tom 14
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nr 2
233-240
EN
We introduce ternary wavelets, based on an interpolating 4-point C^2 ternary stationary subdivision scheme, for compressing fractal-like signals. These wavelets are tightly squeezed and therefore they are more suitable for compressing fractal-like signals. The error in compressing fractal-like signals by ternary wavelets is at most half of that given by four-point wavelets (Wei and Chen, 2002). However, for compressing regular signals we further classify ternary wavelets into 'odd ternary' and 'even ternary' wavelets. Our odd ternary wavelets are better in part for compressing both regular and fractal-like signals than four-point wavelets. These ternary wavelets are locally supported, symmetric and stable. The analysis and synthesis algorithms have linear time complexity.
2
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Application of cubic box spline wavelets in the analysis of signal singularities

100%
Rakowski W.
International Journal of Applied Mathematics and Computer Science
|
2015
|
tom 25
|
nr 4
927-941
EN
In the subject literature, wavelets such as the Mexican hat (the second derivative of a Gaussian) or the quadratic box spline are commonly used for the task of singularity detection. The disadvantage of the Mexican hat, however, is its unlimited support; the disadvantage of the quadratic box spline is a phase shift introduced by the wavelet, making it difficult to locate singular points. The paper deals with the construction and properties of wavelets in the form of cubic box splines which have compact and short support and which do not introduce a phase shift. The digital filters associated with cubic box wavelets that are applied in implementing the discrete dyadic wavelet transform are defined. The filters and the algorithme à trous of the discrete dyadic wavelet transform are used in detecting signal singularities and in calculating the measures of signal singularities in the form of a Lipschitz exponent. The article presents examples illustrating the use of cubic box spline wavelets in the analysis of signal singularities.
3
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Maximal functions and smoothness spaces in $L_{p}(ℝ^{d})

100%
Kyriazis G. C.
Studia Mathematica
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1998
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tom 128
|
nr 3
219-241
EN
We study smoothness spaces generated by maximal functions related to the local approximation errors of integral operators. It turns out that in certain cases these smoothness classes coincide with the spaces $C^α_p(ℝ^d)$, 0 < p≤∞, introduced by DeVore and Sharpley [DS] by means of the so-called sharp maximal functions of Calderón and Scott. As an application we characterize the $C^α_p(ℝ^d)$ spaces in terms of the coefficients of wavelet decompositions.
4
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Ondelettes et poids de Muckenhoupt

100%
Gilles Lemarié-Rieusset P.
Studia Mathematica
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1994
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tom 108
|
nr 2
127-147
EN
We study, for a basis of Hölderian compactly supported wavelets, the boundedness and convergence of the associated projectors $P_j$ on the space $L^p(dμ)$ for some p in ]1,∞[ and some nonnegative Borel measure μ on ℝ. We show that the convergence properties are related to the $A_p$ criterion of Muckenhoupt.
5
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Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces

88%
García-Cuerva J., Kazarian K. S.
Studia Mathematica
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1994
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tom 109
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nr 3
255-276
EN
We study sufficient conditions on the weight w, in terms of membership in the $A_p$ classes, for the spline wavelet systems to be unconditional bases of the weighted space $H^p(w)$. The main tool to obtain these results is a very simple theory of regular Calderón-Zygmund operators.
6
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Approximation properties of wavelets and relations among scaling moments II

75%
Finěk V.
Open Mathematics
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2004
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tom 2
|
nr 4
605-613
EN
A new orthonormality condition for scaling functions is derived. This condition shows a close connection between orthonormality and relations among discrete scaling moments. This new condition in connection with certain approximation properties of scaling functions enables to prove new relations among discrete scaling moments and consequently the same relations for continuous scaling moments.
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