Difference between revisions of "Dirichlet series for an analytic almost-periodic function"
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| + | $#C+1 = 27 : ~/encyclopedia/old_files/data/D032/D.0302930 Dirichlet series for an analytic almost\AAhperiodic function | ||
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A series of the type | 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 | + | 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 | + | 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 | ||
| − | < | + | $$ |
| + | \sum _ {\Lambda _ {n} < 0 } A _ {n} e ^ {\Lambda _ {n} s } ,\ \ | ||
| + | \sum _ {\Lambda _ {n} \geq 0 } A _ {n} e ^ {\Lambda _ {n} s } | ||
| + | $$ | ||
| − | are the Dirichlet series of two functions | + | 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=17761
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=17761
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article