arXiv subjects

Alan J. Pryde

Publications and source records attributed to Alan J. Pryde.

3 recordsLinked to original sources

arXiv2012

Equality of uniform and Carleman spectra for bounded measurable functions

In this paper we study various types of spectra of functions $ϕ:\jj\to X$, where $\jj\in\{\r_+,\r\}$ and $X$ is a complex Banach space. We show that uniform spectrum defined in [15] coincides with Carleman spectrum for $ϕ\in L^{\infty}(\r,X)$. This result holds true also for Laplace (half-line) spectrum for $ϕ\in L^{\infty}(\r_+,X)$. We also indicate a class of bounded measurable functions for which Laplace spectrum and Carleman spectrum are equal

math.FA↗

arXiv2011

Comparison of spectra of absolutely regular distributions and applications

We study the reduced Beurling spectra $sp_{\Cal {A},V} (F)$ of functions $F \in L^1_{loc} (\jj,X)$ relative to certain function spaces $\Cal{A}\st L^{\infty}(\jj,X)$ and $V\st L^1 (\r)$ 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'_{ar}(\jj,X)$ of absolutely regular distributions and has uniformly continuous indefinite integral with $0\not\in sp_{\A,\f(\r)} (F)$ (for example if F is slowly oscillating and $\A$ is $\{0\}$ or $C_0 (\jj,X)$), then $F$ is ergodic. If $F\in \f'_{ar}(\r,X)$ and $M_h F (\cdot)= \int_0^h F(\cdot+s)\,ds$ is bounded for all $h > 0$ (for example if $F$ is ergodic) and if $sp_{C_0(\r,X),\f} (F)=\emptyset$, then ${F}*ψ\in C_0(\r,X)$ for all $ψ\in \f(\r)$. 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

math.FA↗

arXiv2010

Spectral characterization of absolutely regular vector-valued distributions

We study the reduced Beurling spectra $sp_{\Cal {A},V} (F)$ of functions $F \in L^1_{loc} (\jj,X)$ relative to certain function spaces $\Cal{A}\st L^{\infty}(\jj,X)$ and $V\st L^1 (\r)$, where $\jj$ is $\r_+$ or $\r$ and $X$ is a Banach space. We show that if $F$ is bounded or slowly oscillating on $\jj$ with $0\not\in sp_{\A,\f} (F)$, where $\A$ is $\{0\}$ or $C_0 (\jj,X)$ for example and $\f=\f(\r)$, then $F$ is ergodic. This result is new even for $F\in BUC(\jj,X)$ and $\A= C_0(\jj,X)$. If $F$ is ergodic and belongs to the space $ \f'_{ar}(\jj,X)$ of absolutely regular distributions and if $sp_{C_0(\jj,X),\f} (F)=\emptyset$, then $\frak{F}*ψ\in C_0(\r,X)$ for all $ψ\in \f(\r)$. Here, $\frak{F}|\jj =F$ and $\frak{F}|(\r\setminus\jj) =0$. We show that tauberian theorems for Laplace transforms follow from results about the reduced spectrum. Our results are more widely applicable than those of previous authors. We demonstrate this and the sharpness of our results through examples.

math.FA↗