Boundary Values and Convolution in Ultradistribution Spaces
World Scientific, 2007 - Mathematics - 217 pages
This book provides the construction and characterization of important ultradistribution spaces and studies properties and calculations of ultradistributions such as boundedness and convolution. Integral transforms of ultradistributions are constructed and analyzed. The general theory of the representation of ultradistributions as boundary values of analytic functions is obtained and the recovery of the analytic functions as Cauchy, Fourier-Laplace, and Poisson integrals associated with the boundary value is proved.Ultradistributions are useful in applications in quantum field theory, partial differential equations, convolution equations, harmonic analysis, pseudo-differential theory, time-frequency analysis, and other areas of analysis. Thus this book is of interest to users of ultradistributions in applications as well as to research mathematicians in areas of analysis.
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Cones in Rn and Kernels
Ultradifferentiable Functions and Ultradistributions
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analytic function arbitrary assertion Banach spaces boundary value results bounded set Cauchy and Poisson Cauchy integral Cauchy kernel class Mp compact subcone cone in Rn convex cone convolution of ultradistributions Corollary defined in 2.9 Definition denote dual cone elements equicontinuous equivalence exists a constant fixed follows formula Fubini's theorem function g G l,oo G Rn G Tc given Hence Hilbert transform holds implies ind lim inverse Fourier transform Lemma linear mapping measurable function n-dimensional n-rant cone n-tuple nondecreasing nonnegative integers norms notation obtain open convex cone Poisson kernel polygonal cone positive constants pr(C proj lim proof is complete proof of Lemma proof of Theorem properties prove regular cone respectively S^Mp satisfies conditions M.1 satisfies M.1 Section sequence Mp singular integral operators smooth functions subset supp symbol tempered distributions tempered ultradistributions topology ultradistribution spaces ultrapolynomial variable