Abstract
We study the reduced Beurling spectra $sp_{{\cal A},V}(F)$ of functions $F \in L^1_{\rm loc} (\jj,X)$ relative to certain function spaces ${\cal A}\subset L^{\infty}(\jj,X)$ and $V\subset L^1 (\r)$ 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^{\p}_{ar}(\r,X)$ of absolutely regular distributions and has uniformly continuous indefinite integral with $0\notin sp_{\A,\f(\r)} (F)$ (for example, if F is slowly oscillating and $\A$ is $\{0\}$ or $C_0 (\jj,X)$), then $F$ is ergodic. If $F\in \f^{\p}_{ar}(\r,X)$ and $M_h F (\cdot)= (1/h) \int_0^h F(\cdot+s)\,ds$ is bounded for all $h > 0$ (for example, if $F$ is ergodic) and if $sp_{C_0(\r,X),\f} (F)=\emptyset$, then ${F}*\psi \in C_0(\r,X)$ for all $\psi\in \f(\r)$. 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.
Citation
Bolis Basit. Alan J. Pryde. "Comparison of spectra of absolutely regular distributions and applications." Rocky Mountain J. Math. 44 (2) 367 - 395, 2014. https://doi.org/10.1216/RMJ-2014-44-2-367
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