Namespaces
Variants
Actions

Difference between revisions of "Dirichlet series for an analytic almost-periodic function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
d0329301.png
 +
$#A+1 = 27 n = 0
 +
$#C+1 = 27 : ~/encyclopedia/old_files/data/D032/D.0302930 Dirichlet series for an analytic almost\AAhperiodic function
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
A series of the type
 
A series of the type
  
<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/d/d032/d032930/d0329301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
f ( s) \sim  \sum _ { n } A _ {n} e ^ {\Lambda _ {n} \tau }
 +
e ^ {i \Lambda _ {n} t }  = \
 +
\sum _ { n } A _ {n} e ^ {\Lambda _ {n} s } ,\  \alpha < \tau < \beta ,
 +
$$
  
representing in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d0329302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d0329303.png" />, the complete Fourier series of the analytic, regular almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d0329304.png" />, defined on the union of straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d0329305.png" /> (cf. [[Almost-periodic analytic function|Almost-periodic analytic function]]).
+
representing in the strip $  ( \alpha , \beta ) $,  
 +
$  - \infty \leq  \alpha < \beta \leq  + \infty $,  
 +
the complete Fourier series of the analytic, regular almost-periodic function $  f ( s) = f ( \tau + it ) $,  
 +
defined on the union of straight lines $  \mathop{\rm Re} ( s) = \tau $(
 +
cf. [[Almost-periodic analytic function|Almost-periodic analytic function]]).
  
To two different almost-periodic functions in the same strip correspond two different Dirichlet series. In the case of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d0329306.png" />-periodic function the series (*) becomes a Laurent series. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d0329307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d0329308.png" /> are known, respectively, as the Dirichlet coefficients and exponents. Unlike for classical Dirichlet series, the set of real exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d0329309.png" /> in (*) may have finite limit points and may even be everywhere dense. If all Dirichlet exponents have the same sign, for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293010.png" /> is an almost-periodic function in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293011.png" /> and if in (*) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293013.png" /> is an almost-periodic function in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293014.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293015.png" /> uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293016.png" />. A similar theorem is valid for positive Dirichlet exponents [[#References|[2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293017.png" /> is an almost-periodic function in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293018.png" /> and if the indefinite integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293019.png" /> in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293020.png" /> is bounded, then the series
+
To two different almost-periodic functions in the same strip correspond two different Dirichlet series. In the case of a $  2 \pi $-
 +
periodic function the series (*) becomes a Laurent series. The numbers $  A _ {n} $
 +
and $  \Lambda _ {n} $
 +
are known, respectively, as the Dirichlet coefficients and exponents. Unlike for classical Dirichlet series, the set of real exponents $  \Lambda _ {n} $
 +
in (*) may have finite limit points and may even be everywhere dense. If all Dirichlet exponents have the same sign, for example, if $  f ( s) $
 +
is an almost-periodic function in a strip $  ( \alpha , \beta ) $
 +
and if in (*) $  \Lambda _ {n} < 0 $,  
 +
then $  f ( s) $
 +
is an almost-periodic function in the strip $  ( \alpha , + \infty ) $,  
 +
and $  \lim\limits _ {\tau \rightarrow + \infty }  f ( s) = 0 $
 +
uniformly with respect to $  t $.  
 +
A similar theorem is valid for positive Dirichlet exponents [[#References|[2]]]. If $  f ( s) $
 +
is an almost-periodic function in a strip $  [ \alpha , \beta ] $
 +
and if the indefinite integral of $  f ( s) $
 +
in the strip $  [ \alpha , \beta ] $
 +
is bounded, then the 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/d/d032/d032930/d03293021.png" /></td> </tr></table>
+
$$
 +
\sum _ {\Lambda _ {n} < 0 } A _ {n} e ^ {\Lambda _ {n} s } ,\ \
 +
\sum _ {\Lambda _ {n} \geq  0 } A _ {n} e ^ {\Lambda _ {n} s }
 +
$$
  
are the Dirichlet series of two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293023.png" /> which are almost-periodic in every strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293025.png" /> or, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032930/d03293027.png" />.
+
are the Dirichlet series of two functions $  f _ {1} ( s) $
 +
and $  f _ {2} ( s) $
 +
which are almost-periodic in every strip $  [ \alpha _ {1} , + \infty ) $,  
 +
$  \alpha _ {1} > \alpha $
 +
or, respectively, $  ( - \infty , \beta _ {1} ] $,  
 +
$  \beta _ {1} < \beta $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,  "Almost periodic functions" , Chelsea, reprint  (1947)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.M. Levitan,  "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,  "Almost periodic functions" , Chelsea, reprint  (1947)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.M. Levitan,  "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,  "Almost periodic functions" , Cambridge Univ. Press  (1932)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Corduneanu,  "Almost periodic functions" , Wiley  (1968)  (Translated from Rumanian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,  "Almost periodic functions" , Cambridge Univ. Press  (1932)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Corduneanu,  "Almost periodic functions" , Wiley  (1968)  (Translated from Rumanian)</TD></TR></table>

Latest revision as of 19:35, 5 June 2020


A series of the type

$$ \tag{* } f ( s) \sim \sum _ { n } A _ {n} e ^ {\Lambda _ {n} \tau } e ^ {i \Lambda _ {n} t } = \ \sum _ { n } A _ {n} e ^ {\Lambda _ {n} s } ,\ \alpha < \tau < \beta , $$

representing in the strip $ ( \alpha , \beta ) $, $ - \infty \leq \alpha < \beta \leq + \infty $, the complete Fourier series of the analytic, regular almost-periodic function $ f ( s) = f ( \tau + it ) $, defined on the union of straight lines $ \mathop{\rm Re} ( s) = \tau $( cf. Almost-periodic analytic function).

To two different almost-periodic functions in the same strip correspond two different Dirichlet series. In the case of a $ 2 \pi $- periodic function the series (*) becomes a Laurent series. The numbers $ A _ {n} $ and $ \Lambda _ {n} $ are known, respectively, as the Dirichlet coefficients and exponents. Unlike for classical Dirichlet series, the set of real exponents $ \Lambda _ {n} $ in (*) may have finite limit points and may even be everywhere dense. If all Dirichlet exponents have the same sign, for example, if $ f ( s) $ is an almost-periodic function in a strip $ ( \alpha , \beta ) $ and if in (*) $ \Lambda _ {n} < 0 $, then $ f ( s) $ is an almost-periodic function in the strip $ ( \alpha , + \infty ) $, and $ \lim\limits _ {\tau \rightarrow + \infty } f ( s) = 0 $ uniformly with respect to $ t $. A similar theorem is valid for positive Dirichlet exponents [2]. If $ f ( s) $ is an almost-periodic function in a strip $ [ \alpha , \beta ] $ and if the indefinite integral of $ f ( s) $ in the strip $ [ \alpha , \beta ] $ is bounded, then the series

$$ \sum _ {\Lambda _ {n} < 0 } A _ {n} e ^ {\Lambda _ {n} s } ,\ \ \sum _ {\Lambda _ {n} \geq 0 } A _ {n} e ^ {\Lambda _ {n} s } $$

are the Dirichlet series of two functions $ f _ {1} ( s) $ and $ f _ {2} ( s) $ which are almost-periodic in every strip $ [ \alpha _ {1} , + \infty ) $, $ \alpha _ {1} > \alpha $ or, respectively, $ ( - \infty , \beta _ {1} ] $, $ \beta _ {1} < \beta $.

References

[1] H. Bohr, "Almost periodic functions" , Chelsea, reprint (1947) (Translated from German)
[2] B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)

Comments

References

[a1] A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932)
[a2] C. Corduneanu, "Almost periodic functions" , Wiley (1968) (Translated from Rumanian)
How to Cite This Entry:
Dirichlet series for an analytic almost-periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_series_for_an_analytic_almost-periodic_function&oldid=46724
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article