论文标题
矢量值多项式，指数多项式和矢量值谐波分析
Vector valued polynomials, exponential polynomials and vector valued harmonic analysis
论文作者
论文摘要
让$ g $是带有单位的拓扑亚伯式半群，让$ e $为banach空间，让$ c（g，e）$表示连续功能$ f \ colon g \ to e $。如果有$ n \ ge 0 $，则c（g，e）$中的函数$ f \是一个通用的多项式，如果每个$ n \ ge ge 0 $，以至于每个$δ_{h_1} \ldotsΔ_{h_ {h_ {n+1}} f = 0 $对于每个$ h_1，h_1，\ ldots，h_ ldots，h_ fldots，h_ fldots，h_ {n+1} $ restor。我们说，如果是多项式，则c（g，e）$中的$ f \是多项式，其翻译的线性跨度为有限的尺寸； $ f $是一个w-u \ circ f $，对于每个$ u \ e^*$中的每个$ u \，而$ f $是本地多项式，如果它是每个有限生成的subsemigroup，则是本地多项式。我们表明，每个类别的多项式，W-多项式，广义多项式，局部多项式都包含在下一类中。如果$ g $是一个阿贝尔集团，并且具有一个密集的亚组，则具有有限的扭转等级，那么这些班级是一致的。我们还介绍了指数多项式的类别和W-Expo \ - 性多项式，并与多项式和W-PolyNomials建立其表示并建立联系。我们还研究了$ c（g，e）$类中的光谱合成和分析。众所周知，如果$ g $是一个紧凑的阿贝利安集团，而$ e $是一个巴拉克空间，那么频谱合成在$ c（g，e）$中。另一方面，我们表明，如果$ g $是一个无限且离散的阿贝尔集团，而$ e $是无限尺寸的Banach空间，那么即使是光谱分析在$ c（g，e）$中也会失败。但是，如果$ g $是离散的，则具有有限的扭转等级，并且如果$ e $是有限尺寸的Banach空间，那么Spectral Synthesis在$ C（g，e）$中成立。
Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $f\colon G\to E$. A function $f\in C(G,E)$ is a generalized polynomial, if there is an $n\ge 0$ such that $Δ_{h_1} \ldots Δ_{h_{n+1}} f=0$ for every $h_1 ,\ldots , h_{n+1} \in G$, where $Δ_h$ is the difference operator. We say that $f\in C(G,E)$ is a polynomial, if it is a generalized polynomial, and the linear span of its translates is of finite dimension; $f$ is a w-polynomial, if $u\circ f$ is a polynomial for every $u\in E^*$, and $f$ is a local polynomial, if it is a polynomial on every finitely generated subsemigroup. We show that each of the classes of polynomials, w-polynomials, generalized polynomials, local polynomials is contained in the next class. If $G$ is an Abelian group and has a dense subgroup with finite torsion free rank, then these classes coincide. We introduce the classes of exponential polynomials and w-expo\-nential polynomials as well, establish their representations and connection with polynomials and w-polynomials. We also investigate spectral synthesis and analysis in the class $C(G,E)$. It is known that if $G$ is a compact Abelian group and $E$ is a Banach space, then spectral synthesis holds in $C(G,E)$. On the other hand, we show that if $G$ is an infinite and discrete Abelian group and $E$ is a Banach space of infinite dimension, then even spectral analysis fails in $C(G,E)$. If, however, $G$ is discrete, has finite torsion free rank and if $E$ is a Banach space of finite dimension, then spectral synthesis holds in $C(G,E)$.
