Singular Integral Equations

Estrada, Ricardo; Kanwal, Ram P.

doi:10.1007/978-1-4612-1382-6

Cited by 115 publications

(86 citation statements)

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“…We want to make some comments about extension of distributions initially defined in R 0 to R. Observe that this problem is of vital importance for renormalization procedures in quantum field theory ( [2,24,49,50]). It also has much relevance to the study of singular integral equations on spaces of distributions [14]. For simplicity, we discuss the problem of extending a distribution from R + to R.…”

Section: Definitionmentioning

confidence: 99%

Tauberian Theorems for the Wavelet Transform

Vindas

Pilipović

Rakić

2010

J Fourier Anal Appl

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Abstract. We make a complete wavelet analysis of asymptotic properties of distributions. The study is carried out via Abelian and Tauberian type results, connecting the boundary asymptotic behavior of the wavelet transform with local and non-local quasiasymptotic properties of elements in the Schwartz class of tempered distributions. Our Tauberian theorems are full characterizations of such asymptotic properties. We also provide precise wavelet characterizations of the asymptotic behavior of elements in the dual of the space of highly time-frequency localized functions over the real line. For the use of the wavelet transform in local analysis, we study the problem of extensions of distributions initially defined on R \ {0} to R; in this extension problem, we explore the asymptotic properties of extensions of a distribution having a prescribed asymptotic behavior. Our results imply intrinsic properties of functions and measures as well, for example, we give a new proof of the classical Littlewood Tauberian theorem for power series.

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Section: Definitionmentioning

confidence: 99%

Tauberian Theorems for the Wavelet Transform

Vindas

Pilipović

Rakić

2010

J Fourier Anal Appl

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“…For certain distributions one can talk about a unique Hilbert transform [9], sayf, in such a case one may take σ = if . Actually, this will be done in Section 6 for the case of periodic distributions.…”

Section: Determination Of Jumps Of Distributions %mentioning

confidence: 99%

Determination of jumps of distributions by differentiated means

Estrada

Vindas

2009

Acta Math Hung

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Differential means are defined in order to find formulas for jumps of distributions. We analyze two types of jumps occurring in the notions of distributed jump behavior and symmetric jump behavior. We start by defining what we call Riesz differentiated means for numerical series, then the differentiated means are extended to distributed evaluations for the Schwartz class of tempered distributions. The jumps of tempered distributions are completely determined by the differentiated means of the Fourier transform. We also find formulas for the jumps in terms of the asymptotic behavior of partial derivatives of harmonic representations and harmonic conjugate functions. Applications to Fourier series are given

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“…(0.1), was considered in [3,4]. According to [3, p. 194], the solution of this equation is given by the formula…”

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confidence: 99%