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''of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l0571102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l0571104.png" />-system''
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An orthonormal system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l0571105.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l0571106.png" /> such that if the series
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{{TEX|auto}}
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{{TEX|done}}
  
<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/l/l057/l057110/l0571107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
''of order  $  p> 2 $,
 +
$  S _ {p} $-
 +
system''
  
converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l0571108.png" />, then its sum belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l0571109.png" />. If the system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711010.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711011.png" />-system for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711012.png" />, it is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711014.png" />-system. S. Banach proved (see [[#References|[2]]]) that from any sequence of functions bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711015.png" /> and orthonormal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711016.png" /> one can extract an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711017.png" />-system. For an orthonormal system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711018.png" /> to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711019.png" />-system it is necessary and sufficient that there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711020.png" /> depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711021.png" /> and such that
+
An orthonormal system of functions $  \{ \phi _ {n} \} _ {n=} 1  ^  \infty  $
 +
of the space  $  L _ {p} $
 +
such that if 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/l/l057/l057110/l05711022.png" /></td> </tr></table>
+
$$ \tag{* }
 +
\sum _ { n= } 1 ^  \infty  a _ {n} \phi _ {n}  $$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711025.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711026.png" />-system for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711027.png" />, then there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711028.png" /> such that
+
converges in  $  L _ {2} $,
 +
then its sum belongs to  $  L _ {p} $.  
 +
If the system of functions  $  \{ \phi _ {n} \} $
 +
is an  $  S _ {p} $-
 +
system for any  $  p > 2 $,
 +
it is called an $  S _  \infty  $-
 +
system. S. Banach proved (see [[#References|[2]]]) that from any sequence of functions bounded in  $  L _ {p} $
 +
and orthonormal in  $  L _ {2} $
 +
one can extract an  $  S _ {p} $-
 +
system. For an orthonormal system of functions  $  \{ \phi _ {n} \} $
 +
to be an  $  S _ {p} $-
 +
system it is necessary and sufficient that there is a constant $  \mu _ {p} $
 +
depending only on  $  p $
 +
and such that
  
<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/l/l057/l057110/l05711029.png" /></td> </tr></table>
+
$$
 +
\left \| \sum _ { n= } 1 ^ { N }  a _ {n} \phi _ {n} \right \| _ {L _ {p}  }  \leq  \
 +
\mu _ {p}  \left \| \sum _ { n= } 1 ^ { N }  a _ {n} \phi _ {n} \right \| _ {L _ {2}  }
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711031.png" />. A system of functions with this property is called a Banach system. These definitions extend to non-orthogonal systems of functions (see [[#References|[3]]], for example). Sometimes a lacunary system of functions is understood to be a system of functions whose series have one or several properties of [[Lacunary trigonometric series|lacunary trigonometric series]], in dependence on which they take different names. For example, with the theory of uniqueness for lacunary trigonometric series there is associated the concept of a lacunary system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711033.png" />-uniqueness. A system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711034.png" /> is called a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711035.png" />-uniqueness if there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711036.png" /> such that the convergence of the series (*) to zero everywhere, except possibly on a set of measure less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057110/l05711037.png" />, implies that all its coefficients are zero.
+
for all $  N $
 +
and  $  \{ a _ {n} \} $.  
 +
If  $  \{ \phi _ {n} \} $
 +
is an  $  S _ {p} $-
 +
system for some  $  p > 2 $,
 +
then there is a constant  $  m $
 +
such that
 +
 
 +
$$
 +
\left \| \sum _ { n= } 1 ^ { N }  a _ {n} \phi _ {n} \right \| _ {L _ {2}  }  \leq  \
 +
m  \left \| \sum _ { n= } 1 ^ { N }  a _ {n} \phi _ {n} \right \| _ {L _ {p}  }
 +
$$
 +
 
 +
for all  $  N $
 +
and  $  \{ a _ {n} \} $.  
 +
A system of functions with this property is called a Banach system. These definitions extend to non-orthogonal systems of functions (see [[#References|[3]]], for example). Sometimes a lacunary system of functions is understood to be a system of functions whose series have one or several properties of [[Lacunary trigonometric series|lacunary trigonometric series]], in dependence on which they take different names. For example, with the theory of uniqueness for lacunary trigonometric series there is associated the concept of a lacunary system of $  \epsilon $-
 +
uniqueness. A system $  \{ \phi _ {n} \} _ {n=} 1  ^  \infty  $
 +
is called a system of $  \epsilon $-
 +
uniqueness if there is a number $  \epsilon > 0 $
 +
such that the convergence of the series (*) to zero everywhere, except possibly on a set of measure less than $  \epsilon $,  
 +
implies that all its coefficients are zero.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Alexits,  "Konvergenzprobleme der Orthogonalreihen" , Deutsch. Verlag Wissenschaft.  (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.F. Gaposhkin,  "Lacunary series and independent functions"  ''Russian Math. Surveys'' , '''21''' :  6  (1966)  pp. 1–82  ''Uspekhi Mat. Nauk'' , '''21''' :  6  (1966)  pp. 3–82</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Alexits,  "Konvergenzprobleme der Orthogonalreihen" , Deutsch. Verlag Wissenschaft.  (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.F. Gaposhkin,  "Lacunary series and independent functions"  ''Russian Math. Surveys'' , '''21''' :  6  (1966)  pp. 1–82  ''Uspekhi Mat. Nauk'' , '''21''' :  6  (1966)  pp. 3–82</TD></TR></table>

Revision as of 22:15, 5 June 2020


of order $ p> 2 $, $ S _ {p} $- system

An orthonormal system of functions $ \{ \phi _ {n} \} _ {n=} 1 ^ \infty $ of the space $ L _ {p} $ such that if the series

$$ \tag{* } \sum _ { n= } 1 ^ \infty a _ {n} \phi _ {n} $$

converges in $ L _ {2} $, then its sum belongs to $ L _ {p} $. If the system of functions $ \{ \phi _ {n} \} $ is an $ S _ {p} $- system for any $ p > 2 $, it is called an $ S _ \infty $- system. S. Banach proved (see [2]) that from any sequence of functions bounded in $ L _ {p} $ and orthonormal in $ L _ {2} $ one can extract an $ S _ {p} $- system. For an orthonormal system of functions $ \{ \phi _ {n} \} $ to be an $ S _ {p} $- system it is necessary and sufficient that there is a constant $ \mu _ {p} $ depending only on $ p $ and such that

$$ \left \| \sum _ { n= } 1 ^ { N } a _ {n} \phi _ {n} \right \| _ {L _ {p} } \leq \ \mu _ {p} \left \| \sum _ { n= } 1 ^ { N } a _ {n} \phi _ {n} \right \| _ {L _ {2} } $$

for all $ N $ and $ \{ a _ {n} \} $. If $ \{ \phi _ {n} \} $ is an $ S _ {p} $- system for some $ p > 2 $, then there is a constant $ m $ such that

$$ \left \| \sum _ { n= } 1 ^ { N } a _ {n} \phi _ {n} \right \| _ {L _ {2} } \leq \ m \left \| \sum _ { n= } 1 ^ { N } a _ {n} \phi _ {n} \right \| _ {L _ {p} } $$

for all $ N $ and $ \{ a _ {n} \} $. A system of functions with this property is called a Banach system. These definitions extend to non-orthogonal systems of functions (see [3], for example). Sometimes a lacunary system of functions is understood to be a system of functions whose series have one or several properties of lacunary trigonometric series, in dependence on which they take different names. For example, with the theory of uniqueness for lacunary trigonometric series there is associated the concept of a lacunary system of $ \epsilon $- uniqueness. A system $ \{ \phi _ {n} \} _ {n=} 1 ^ \infty $ is called a system of $ \epsilon $- uniqueness if there is a number $ \epsilon > 0 $ such that the convergence of the series (*) to zero everywhere, except possibly on a set of measure less than $ \epsilon $, implies that all its coefficients are zero.

References

[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[2] G. Alexits, "Konvergenzprobleme der Orthogonalreihen" , Deutsch. Verlag Wissenschaft. (1960)
[3] V.F. Gaposhkin, "Lacunary series and independent functions" Russian Math. Surveys , 21 : 6 (1966) pp. 1–82 Uspekhi Mat. Nauk , 21 : 6 (1966) pp. 3–82
How to Cite This Entry:
Lacunary system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lacunary_system&oldid=47551
This article was adapted from an original article by V.F. Emel'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article