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A class of almost-periodic functions in which the analogue of the Riesz–Fischer theorem is valid: Any trigonometric series
 
A class of almost-periodic functions in which the analogue of the Riesz–Fischer theorem is valid: Any trigonometric 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/b/b015/b015820/b0158201.png" /></td> </tr></table>
+
$$
 +
\sum _ { n } a _ {n} e ^ {i \lambda _ {n} x } ,
 +
$$
  
 
where
 
where
  
<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/b/b015/b015820/b0158202.png" /></td> </tr></table>
+
$$
 +
\sum _ { n } | a _ {n} |  ^ {2}  < \infty ,
 +
$$
  
is the Fourier series of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b0158203.png" />-almost-periodic function. The definition of these functions [[#References|[1]]], [[#References|[2]]] is based on a generalization of the concept of an [[Almost-period|almost-period]], and certain additional ideas must be introduced in it. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b0158204.png" /> of real numbers is called sufficiently homogeneous if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b0158205.png" /> such that the ratio between the largest number of members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b0158206.png" /> in an interval of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b0158207.png" /> and the smallest number of members in an interval of the same length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b0158208.png" /> is less than 2. A sufficiently homogeneous set is also relatively dense. A complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b0158209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582010.png" />, summable to degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582011.png" /> on any finite interval of the real axis, is called a Besicovitch almost-periodic function if to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582012.png" /> there corresponds a sufficiently homogeneous set of numbers (the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582013.png" />-almost-periods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582014.png" />):
+
is the Fourier series of some $  B  ^ {2} $-
 +
almost-periodic function. The definition of these functions [[#References|[1]]], [[#References|[2]]] is based on a generalization of the concept of an [[Almost-period|almost-period]], and certain additional ideas must be introduced in it. A set $  E $
 +
of real numbers is called sufficiently homogeneous if there exists an $  L > 0 $
 +
such that the ratio between the largest number of members of $  E $
 +
in an interval of length $  L $
 +
and the smallest number of members in an interval of the same length $  L $
 +
is less than 2. A sufficiently homogeneous set is also relatively dense. A complex-valued function $  f(x) $,
 +
$  - \infty < x < \infty $,  
 +
summable to degree $  p $
 +
on any finite interval of the real axis, is called a Besicovitch almost-periodic function if to each $  \epsilon > 0 $
 +
there corresponds a sufficiently homogeneous set of numbers (the so-called $  (B  ^ {p} , \epsilon ) $-
 +
almost-periods of $  f(x) $):
  
<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/b/b015/b015820/b01582015.png" /></td> </tr></table>
+
$$
 +
{} \dots < \tau _ {-2}  < \tau _ {-1}  < \tau _ {0< \tau _ {1}  < \dots ,
 +
$$
  
such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582016.png" />
+
such that for each $  i $
  
<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/b/b015/b015820/b01582017.png" /></td> </tr></table>
+
$$
 +
\overline{M}\; _ {x} \{ | f (x + \tau _ {i} ) - f (x) |  ^ {p} \}
 +
< \epsilon  ^ {p} ,
 +
$$
  
and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582018.png" />
+
and for each $  c > 0 $
  
<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/b/b015/b015820/b01582019.png" /></td> </tr></table>
+
$$
 +
\overline{M}\; _ {x} \overline{M}\; _ {i} {
 +
\frac{1}{c}
 +
}
 +
\int\limits _ { x } ^ { x+c }  | f ( \xi + \tau _ {i} ) -
 +
f (x) |  ^ {p}  d \xi  < \epsilon  ^ {p} ,
 +
$$
  
 
where
 
where
  
<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/b/b015/b015820/b01582020.png" /></td> </tr></table>
+
$$
 +
\overline{M}\; _ {x} \{ F (x) \}  = \
 +
{\lim\limits _ {\tau \rightarrow \infty } } bar \
 +
{
 +
\frac{1}{2 \tau }
 +
}
 +
\int\limits _ {- \tau } ^  \tau 
 +
F (x)  dx,
 +
$$
  
<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/b/b015/b015820/b01582021.png" /></td> </tr></table>
+
$$
 +
\overline{M}\; _ {i} \{ F (i) \}  = {\lim\limits _ {n \rightarrow \infty
 +
} } bar  {
 +
\frac{1}{2n+1}
 +
} \sum _ { i=-n } ^ { n }  F (i).
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582022.png" /> is a real-valued function, defined, respectively, for a real variable and an integer argument.
+
Here $  F(x) $
 +
is a real-valued function, defined, respectively, for a real variable and an integer argument.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.S. Besicovitch,  "On mean values of functions of a complex and of a real variable"  ''Proc. London Math. Soc. (2)'' , '''27'''  (1927)  pp. 373–388</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Besicovitch,  "On Parseval's theorem for Dirichlet series"  ''Proc. London Math. Soc. (2)'' , '''26'''  (1927)  pp. 25–34</TD></TR><TR><TD valign="top">[3]</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">  A.S. Besicovitch,  "On mean values of functions of a complex and of a real variable"  ''Proc. London Math. Soc. (2)'' , '''27'''  (1927)  pp. 373–388</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Besicovitch,  "On Parseval's theorem for Dirichlet series"  ''Proc. London Math. Soc. (2)'' , '''26'''  (1927)  pp. 25–34</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.M. Levitan,  "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Besicovitch developed his theory in [[#References|[a1]]], rather than in [[#References|[1]]], [[#References|[2]]].
 
Besicovitch developed his theory in [[#References|[a1]]], rather than in [[#References|[1]]], [[#References|[2]]].
  
As is implicit in the article, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582023.png" /> there is a class of almost-periodic functions, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582024.png" />. The first part of the article deals with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015820/b01582025.png" />, the rest is more general. General references may be found under [[Almost-periodic function|Almost-periodic function]].
+
As is implicit in the article, for each $  p \geq  1 $
 +
there is a class of almost-periodic functions, denoted by $  B  ^ {p} $.  
 +
The first part of the article deals with $  B  ^ {2} $,  
 +
the rest is more general. General references may be found under [[Almost-periodic function|Almost-periodic function]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,  "On generalized almost periodic functions"  ''Proc. London Math. Soc. (2)'' , '''25'''  (1926)  pp. 495–512</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,  "On generalized almost periodic functions"  ''Proc. London Math. Soc. (2)'' , '''25'''  (1926)  pp. 495–512</TD></TR></table>

Revision as of 10:58, 29 May 2020


A class of almost-periodic functions in which the analogue of the Riesz–Fischer theorem is valid: Any trigonometric series

$$ \sum _ { n } a _ {n} e ^ {i \lambda _ {n} x } , $$

where

$$ \sum _ { n } | a _ {n} | ^ {2} < \infty , $$

is the Fourier series of some $ B ^ {2} $- almost-periodic function. The definition of these functions [1], [2] is based on a generalization of the concept of an almost-period, and certain additional ideas must be introduced in it. A set $ E $ of real numbers is called sufficiently homogeneous if there exists an $ L > 0 $ such that the ratio between the largest number of members of $ E $ in an interval of length $ L $ and the smallest number of members in an interval of the same length $ L $ is less than 2. A sufficiently homogeneous set is also relatively dense. A complex-valued function $ f(x) $, $ - \infty < x < \infty $, summable to degree $ p $ on any finite interval of the real axis, is called a Besicovitch almost-periodic function if to each $ \epsilon > 0 $ there corresponds a sufficiently homogeneous set of numbers (the so-called $ (B ^ {p} , \epsilon ) $- almost-periods of $ f(x) $):

$$ {} \dots < \tau _ {-2} < \tau _ {-1} < \tau _ {0} < \tau _ {1} < \dots , $$

such that for each $ i $

$$ \overline{M}\; _ {x} \{ | f (x + \tau _ {i} ) - f (x) | ^ {p} \} < \epsilon ^ {p} , $$

and for each $ c > 0 $

$$ \overline{M}\; _ {x} \overline{M}\; _ {i} { \frac{1}{c} } \int\limits _ { x } ^ { x+c } | f ( \xi + \tau _ {i} ) - f (x) | ^ {p} d \xi < \epsilon ^ {p} , $$

where

$$ \overline{M}\; _ {x} \{ F (x) \} = \ {\lim\limits _ {\tau \rightarrow \infty } } bar \ { \frac{1}{2 \tau } } \int\limits _ {- \tau } ^ \tau F (x) dx, $$

$$ \overline{M}\; _ {i} \{ F (i) \} = {\lim\limits _ {n \rightarrow \infty } } bar { \frac{1}{2n+1} } \sum _ { i=-n } ^ { n } F (i). $$

Here $ F(x) $ is a real-valued function, defined, respectively, for a real variable and an integer argument.

References

[1] A.S. Besicovitch, "On mean values of functions of a complex and of a real variable" Proc. London Math. Soc. (2) , 27 (1927) pp. 373–388
[2] A.S. Besicovitch, "On Parseval's theorem for Dirichlet series" Proc. London Math. Soc. (2) , 26 (1927) pp. 25–34
[3] B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)

Comments

Besicovitch developed his theory in [a1], rather than in [1], [2].

As is implicit in the article, for each $ p \geq 1 $ there is a class of almost-periodic functions, denoted by $ B ^ {p} $. The first part of the article deals with $ B ^ {2} $, the rest is more general. General references may be found under Almost-periodic function.

References

[a1] A.S. Besicovitch, "On generalized almost periodic functions" Proc. London Math. Soc. (2) , 25 (1926) pp. 495–512
How to Cite This Entry:
Besicovitch almost-periodic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Besicovitch_almost-periodic_functions&oldid=11650
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article