<i>Les Fonctions généralisées ou Distributions</i>

Bouix, Maurice; Mansfield, Janet

doi:10.1063/1.3047553

Cited by 4 publications

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“…The paper is organised as follows. This Section briefly overviews wavelets and some relevant aspects of the Laurent Schwartz theory of distributions [10][11][12], as well as their relationship with wavelets. Section II addresses standard Taylor series viewed as Tukey-tapers [13].…”

Section: Introductionmentioning

confidence: 99%

“…Let T be a distribution of bounded support, thus compact. One can prove [10] that T can be extended to the space D:=C ∞ (R). The set of compactly supported distributions is denoted by D'.…”

Section: Introductionmentioning

confidence: 99%

“…The set of compactly supported distributions is denoted by D'. Continuous wavelet transforms [1,4] can also be interpreted as distributions [10,11]. If ψ(t) is a mother wavelet, then:…”

Section: Introductionmentioning

confidence: 99%

“…The homothetic and translation relationships of distributions [10] correspond to usual scaling and translation in wavelets. i)…”

Section: Introductionmentioning

confidence: 99%

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Taylor Series as Wide-sense Biorthogonal Wavelet Decomposition

Oliveira,

Lins

2005

Anais Do XXII Simpósio Brasileiro De Telecomunicações

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Resumo-Wavelets generalizadas de suporte pontual são definidas com base em impulsos de Dirac, doublet e outras derivadas de δ δ δ δ(t). Uma análise biortogonal lato sensu conduz às séries clássicas de Taylor e novas séries duais interpretadas como distribuições de Laurent Schwartz. Uma identidade isométrica do tipo Parseval é deduzida a partir de séries de Taylor, mostrando que as séries de Taylor também possuem um teorema de energia. Novas representações de sinais denominadas de 'derivagramas', similares aos 'espectrogramas', são introduzidas. Esta abordagem corrobora o impacto da teoria de wavelets na moderna análise de sinais.Palavras-Chave-wavelets de suporte pontual, análise biortogonal, Teoremas de Parseval, derivagramas.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The set of compactly supported distributions is denoted by D'. Continuous wavelet transforms [1,4] can also be interpreted as distributions [10,11]. If ψ(t) is a mother wavelet, then:…”

Section: Introductionmentioning

confidence: 99%

“…The homothetic and translation relationships of distributions [10] correspond to usual scaling and translation in wavelets. i)…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Taylor Series as Wide-sense Biorthogonal Wavelet Decomposition

Oliveira,

Lins

2005

Anais Do XXII Simpósio Brasileiro De Telecomunicações

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The Division of Distributions by the Independent Variable

Berg

1977

Mathematische Nachrichten

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I n the theory of distributions the multiplication of a distribution by a holomorphic function without, singularities is well defined. For instance, we havewhere 6 is DIRAC'S delta distribution. Sometimes, more general products are defined as cf.[l], p. 249). But if one tries to calculate with l/t in a naive way, one gets nequalit,ies like so that there is no associativity.The main result of this paper is the following: It may be arranged, that one has quite well associativity, but no commutativity. All considerat.ions are essentially based on the theory of right invertible operators in the framework of the general operational calculus (cf. {2], [7]). To put things clear, we restrict ourselves at the beginning to a simple subclass of distributions. An isomorphism correslmnding to the LAPLACE transform shows, that our definitions are quite natural. It is easily possible to extend the results to wider classes of distributions by using a H -4 3 1 .~~ basis, converging series with a suitable topology (cf. [9]) or formal series (cf. 161). Such extensions are sketched at the end of the paper. Preliminaries. Let 23 be the least vector space generated by one element b i o and by two linear right invertible operators D, T with (1) DT -TD = 1, TDb = Oin such a way. that D and T map 23 onto 23. The field R of the coefficients of % shall have the characteristic zero. So we can assume, that the field of rational numbers is embeded in R, and that 1 is the unit operator.

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Complex-variable distribution theory for Laplace and z transforms

Corinthios

2005

IEE Proc., Vis. Image Process.

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Les Fonctions généralisées ou Distributions

Cited by 4 publications

References 0 publications

Taylor Series as Wide-sense Biorthogonal Wavelet Decomposition

Taylor Series as Wide-sense Biorthogonal Wavelet Decomposition

The Division of Distributions by the Independent Variable

Complex-variable distribution theory for Laplace and z transforms

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