Comparison of spectra of absolutely regular distributions and applications

2011/08/26 by Bolis Basit, Basit, Bolis, A. J. Pryde +2
Economics, Econometrics and Finance · Engineering · Mathematics · #Advanced Banach Space Theory #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #math.FA

paper · pdf · doi:10.48550/arxiv.1108.5237

24 pages

arxiv created 2011/08/26 · arxiv updated 2011/08/29

Abstract

We study the reduced Beurling spectra sp_\Cal A,V (F) of functions F ∈ L¹_loc (\jj,X) relative to certain function spaces \CalA\st L^∞(\jj,X) and V\st L¹ (\r) and compare them with other spectra including the weak Laplace spectrum. Here \jj is \r₊ or \r and X is a Banach space. If F belongs to the space \f'_ar(\jj,X) of absolutely regular distributions and has uniformly continuous indefinite integral with 0\not∈ sp_\A,\f(\r) (F) (for example if F is slowly oscillating and \A is \0\ or C₀ (\jj,X)), then F is ergodic. If F∈ \f'_ar(\r,X) and M_h F (⋅)= ∫₀^h F(⋅+s) ds is bounded for all h > 0 (for example if F is ergodic) and if sp_C0(\r,X),\f (F)=∅, then F*ψ∈ C₀(\r,X) for all ψ∈ \f(\r). We show that tauberian theorems for Laplace transforms follow from results about reduced spectra. Our results are more general than previous ones and we demonstrate this through examples

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