Namespaces
Variants
Actions

Difference between revisions of "Almost-periodic analytic function"

From Encyclopedia of Mathematics
Jump to: navigation, search
m
m (tex encoding is done)
 
Line 1: Line 1:
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
+
An analytic function $f(s)$, $s=\sigma+i\tau$, regular in a strip $-\infty\leqslant\alpha<\sigma<\beta\leqslant+\infty$, and expandable into a series
 +
\begin{equation}
 +
\sum a_ne^{i\lambda_ns},
 +
\end{equation}
  
<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>
+
where the $a_n$ are complex and the $\lambda_n$ are real numbers. A real number $\tau$ is called an $\varepsilon$-almost-period of $f(s)$ if for all points of the strip $(\alpha, \beta)$ the inequality
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119605.png" /> are complex and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119606.png" /> are real numbers. A real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119607.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119609.png" />-almost-period of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196010.png" /> if for all points of the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196011.png" /> the inequality
+
\begin{equation}
 +
|f(s+i\tau) - f(s)|<\varepsilon
 +
\end{equation}
  
<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/a01196012.png" /></td> </tr></table>
+
holds. An almost-periodic analytic function is an analytic function that is regular in a strip $(\alpha, \beta)$ and possesses a relatively-dense set of $\varepsilon$-almost-periods for every $\varepsilon>0$. An almost-periodic analytic function on a closed strip $\alpha\leqslant\sigma\leqslant\beta$ is defined similarly. An almost-periodic analytic function on a strip $[\alpha, \beta]$ is a uniformly almost-periodic function of the real variable $\tau$ on every straight line in the strip and it is bounded in $[\alpha, \beta]$, i.e. on any interior strip. If a function $f(s)$, regular in a strip $(\alpha, \beta)$, is a uniformly almost-periodic function on at least one line $\sigma=\sigma_0$ in the strip, then boundedness of $f(s)$ in $[\alpha, \beta]$ implies its almost-periodicity on the entire strip $[\alpha, \beta]$. 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|almost-periodic function]]). Therefore, many important results of the latter theory can be easily carried over to almost-periodic analytic functio
 
+
ns: the uniqueness theorem, Parseval's equality, rules of operation with Dirichlet series, the approximation theorem, and several other theorems.
holds. An almost-periodic analytic function is an analytic function that is regular in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196013.png" /> and possesses a relatively-dense set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196015.png" />-almost-periods for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196016.png" />. An almost-periodic analytic function on a closed strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196017.png" /> is defined similarly. An almost-periodic analytic function on a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196018.png" /> is a uniformly almost-periodic function of the real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196019.png" /> on every straight line in the strip and it is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196020.png" />, i.e. on any interior strip. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196021.png" />, regular in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196022.png" />, is a uniformly almost-periodic function on at least one line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196023.png" /> in the strip, then boundedness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196025.png" /> implies its almost-periodicity on the entire strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196026.png" />. 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|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====
 
====References====

Latest revision as of 08:21, 16 October 2013

An analytic function $f(s)$, $s=\sigma+i\tau$, regular in a strip $-\infty\leqslant\alpha<\sigma<\beta\leqslant+\infty$, and expandable into a series \begin{equation} \sum a_ne^{i\lambda_ns}, \end{equation}

where the $a_n$ are complex and the $\lambda_n$ are real numbers. A real number $\tau$ is called an $\varepsilon$-almost-period of $f(s)$ if for all points of the strip $(\alpha, \beta)$ the inequality

\begin{equation} |f(s+i\tau) - f(s)|<\varepsilon \end{equation}

holds. An almost-periodic analytic function is an analytic function that is regular in a strip $(\alpha, \beta)$ and possesses a relatively-dense set of $\varepsilon$-almost-periods for every $\varepsilon>0$. An almost-periodic analytic function on a closed strip $\alpha\leqslant\sigma\leqslant\beta$ is defined similarly. An almost-periodic analytic function on a strip $[\alpha, \beta]$ is a uniformly almost-periodic function of the real variable $\tau$ on every straight line in the strip and it is bounded in $[\alpha, \beta]$, i.e. on any interior strip. If a function $f(s)$, regular in a strip $(\alpha, \beta)$, is a uniformly almost-periodic function on at least one line $\sigma=\sigma_0$ in the strip, then boundedness of $f(s)$ in $[\alpha, \beta]$ implies its almost-periodicity on the entire strip $[\alpha, \beta]$. 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 functio ns: 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
How to Cite This Entry:
Almost-periodic analytic function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Almost-periodic_analytic_function&oldid=30620
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article