Difference between revisions of "Almost-periodic analytic function"
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− | An analytic function | + | An analytic function $f(s)$,<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119602.png" />, regular in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119603.png" />, and expandable into a series |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119604.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119604.png" /></td> </tr></table> |
Revision as of 08:01, 6 March 2013
An analytic function $f(s)$,, regular in a strip
, and expandable into a series
![]() |
where the are complex and the
are real numbers. A real number
is called an
-almost-period of
if for all points of the strip
the inequality
![]() |
holds. An almost-periodic analytic function is an analytic function that is regular in a strip and possesses a relatively-dense set of
-almost-periods for every
. An almost-periodic analytic function on a closed strip
is defined similarly. An almost-periodic analytic function on a strip
is a uniformly almost-periodic function of the real variable
on every straight line in the strip and it is bounded in
, i.e. on any interior strip. If a function
, regular in a strip
, is a uniformly almost-periodic function on at least one line
in the strip, then boundedness of
in
implies its almost-periodicity on the entire strip
. Consequently, the theory of almost-periodic analytic functions turns out to be a theory analogous to that of almost-periodic functions of a real variable (cf. almost-periodic function). Therefore, many important results of the latter theory can be easily carried over to almost-periodic analytic functions: the uniqueness theorem, Parseval's equality, rules of operation with Dirichlet series, the approximation theorem, and several other theorems.
References
[1] | H. Bohr, "Almost-periodic functions" , Chelsea, reprint (1947) (Translated from German) |
[2] | B.M. Levitan, "Almost-periodic functions" , Moscow (1953) pp. Chapt. 7 (In Russian) |
Comments
The hyphen between almost and periodic is sometimes dropped.
References
[a1] | C. Corduneanu, "Almost periodic functions" , Interscience (1961) pp. Chapt. 3 |
Almost-periodic analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-periodic_analytic_function&oldid=29514