Almost-period
A concept from the theory of almost-periodic functions (cf. Almost-periodic function); a generalization of the notion of a period. For a uniformly almost-periodic function ,
, a number
is called an
-almost-period of
if for all
,
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For generalized almost-periodic functions the concept of an almost-period is more complicated. For example, in the space an
-almost-period
is defined by the inequality
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where is the distance between
and
in the metric of
.
A set of almost-periods of a function is said to be relatively dense if there is a number
such that every interval
of the real line contains at least one number from this set. The concepts of uniformly almost-periodic functions and that of Stepanov almost-periodic functions may be defined by requiring the existence of relatively-dense sets of
-almost-periods for these functions.
References
[1] | B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian) |
Comments
For the definition of and its metric
see Almost-periodic function. The Weyl, Besicovitch and Levitan almost-periodic functions can also be characterized in terms of
-periods. These characterizations are more complicated. A good additional reference is [a1], especially Chapt. II.
References
[a1] | A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932) |
Almost-period. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-period&oldid=32494