Almost Automorphic Functions on Groups
Almost Automorphic Functions on Groups
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DOI：
10.2307/2373071
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发表时间：
1965-07
影响因子：
1.7
通讯作者：
W. Veech
中科院分区：
文献类型：
--
作者：
W. Veech
Introduction. S. Bochner has observed in various coiitexts that a certain property enjoyed by the almost periodic functions on a group G canl be used with advantage in obtaininig simpler and conceptually more natural proofs of certain theorems concerninig these functions. ([2], [4], [5].) Bochner calls his property " almiost automorphy " because it first arose in work on differential geometry. Taking G for the present to be the group of integers (= Z) an almost automorphic function f has the property that from any sequence {s'} C Z may be extracted a subsequence {cf} such that both limf(t+ cn) =g(t) and limg(tn) ==f(t) hold for each tC Z and some n -oo t-oo function g, but not necessarily uniformly. Bochner has observed that almost periodic functions are almost automorphic, but the converse is not true. ([5], [18].) However we will show in the present paper that the almost automorphic functiolls on a group can be characterized in terms of the almost periodic functions. A function f on G is almost automorphic if and only if it is the pointwise limit of a " jointly almost automorphic" net of almost periodic functions. (A consequence of this result is that a group is maximally (minimally) almost automorphic if and only if it is maximally (minimally) almost periodic.) Conversely one can characterize almost periodicity in terms, of almost automorphy: A function f is almost periodic if and only if lim f (t + ,) = g (t) is almost automorphic whenever the limit exists. This lb
Introduction. S. Bochner has observed in various coiitexts that a certain property enjoyed by the almost periodic functions on a group G canl be used with advantage in obtaininig simpler and conceptually more natural proofs of certain theorems concerninig these functions. ([2], [4], [5].) Bochner calls his property " almiost automorphy " because it first arose in work on differential geometry. Taking G for the present to be the group of integers (= Z) an almost automorphic function f has the property that from any sequence {s'} C Z may be extracted a subsequence {cf} such that both limf(t+ cn) =g(t) and limg(tn) ==f(t) hold for each tC Z and some n -oo t-oo function g, but not necessarily uniformly. Bochner has observed that almost periodic functions are almost automorphic, but the converse is not true. ([5], [18].) However we will show in the present paper that the almost automorphic functiolls on a group can be characterized in terms of the almost periodic functions. A function f on G is almost automorphic if and only if it is the pointwise limit of a " jointly almost automorphic" net of almost periodic functions. (A consequence of this result is that a group is maximally (minimally) almost automorphic if and only if it is maximally (minimally) almost periodic.) Conversely one can characterize almost periodicity in terms, of almost automorphy: A function f is almost periodic if and only if lim f (t + ,) = g (t) is almost automorphic whenever the limit exists. This lb