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Estimates of the Fourier coefficients of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467501.png" />; established by W.H. Young [[#References|[1]]] and F. Hausdorff [[#References|[2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467502.png" /> be an [[Orthonormal system|orthonormal system]] of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467503.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467504.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467505.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467506.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467508.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h0467509.png" />, then
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<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/h/h046/h046750/h04675010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675011.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675013.png" /> converges, there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675014.png" /> such that
+
Estimates of the Fourier coefficients of functions in  $  L _ {p} $;
 +
established by W.H. Young [[#References|[1]]] and F. Hausdorff [[#References|[2]]]. Let  $  \phi _ {n} $
 +
be an [[Orthonormal system|orthonormal system]] of functions on  $  [ a, b] $,
 +
let  $  | \phi _ {n} ( t) | \leq  M $
 +
for all  $  t \in [ a, b] $
 +
and for all  $  n = 1, 2 \dots $
 +
and let  $  1 < p \leq  2 $,
 +
$  1/p + 1/p  ^  \prime  = 1 $.  
 +
If  $  f \in L _ {p} $,
 +
then
  
<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/h/h046/h046750/h04675015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
\left (
 +
\sum _ {n = 1 } ^  \infty 
 +
| c _ {n} ( f  ) | ^ {p  ^  \prime  }
 +
\right ) ^ {1/p  ^  \prime  }  \leq  \
 +
M ^ {( 2 - p) / p }
 +
\left ( \int\limits _ { a } ^ { b }
 +
| f ( t) |  ^ {p}  dt
 +
\right )  ^ {1/p} ,
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675016.png" /> one may take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675017.png" />, and this series converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675018.png" />.
+
where  $  c _ {n} ( f  ) $
 +
are the Fourier coefficients of  $  f $.  
 +
If  $  \sum _ {n = 1 }  ^  \infty  | a _ {n} |  ^ {p} $
 +
converges, there exists a function  $  g $
 +
such that
  
The Hausdorff–Young inequalities (1) and (2) are equivalent. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675019.png" /> they do not hold. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675020.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675021.png" />, then there exists a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675022.png" /> such that its Fourier coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675023.png" /> in the trigonometric system satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675024.png" />. A qualitative statement of the Hausdorff–Young inequality (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675027.png" />) for unbounded orthonormal systems of functions does not hold, in general. An analogue of the Hausdorff–Young inequalities is valid for a broad class of function spaces.
+
$$ \tag{2 }
 +
\left (
 +
\int\limits _ { a } ^ { b }
 +
| g ( t) | ^ {p  ^  \prime  }
 +
dt \right ) ^ {1/p  ^  \prime  }  \leq  \
 +
M ^ {( 2 - p) / p }
 +
\left ( \sum _ {n = 1 } ^  \infty 
 +
| a _ {n} |  ^ {p}
 +
\right )  ^ {1/p} .
 +
$$
 +
 
 +
For  $  g $
 +
one may take  $  \sum _ {n = 1 }  ^  \infty  a _ {n} \phi _ {n} $,
 +
and this series converges in  $  L _ {p  ^  \prime  } $.
 +
 
 +
The Hausdorff–Young inequalities (1) and (2) are equivalent. For $  p > 2 $
 +
they do not hold. Moreover, if $  b _ {n} \geq  0 $
 +
and if $  \sum _ {n = 1 }  ^  \infty  b _ {n}  ^ {2} < \infty $,  
 +
then there exists a continuous function $  f $
 +
such that its Fourier coefficients $  c _ {n} ( f  ) $
 +
in the trigonometric system satisfy the condition $  | c _ {n} ( f  ) | > b _ {n} $.  
 +
A qualitative statement of the Hausdorff–Young inequality (if $  f \in L _ {p} $,  
 +
$  1 \leq  p \leq  2 $,  
 +
then $  \{ c _ {n} ( f  ) \} \in l _ {p  ^  \prime  } $)  
 +
for unbounded orthonormal systems of functions does not hold, in general. An analogue of the Hausdorff–Young inequalities is valid for a broad class of function spaces.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.H. Young,  "On the determination of the summability of a function by means of its Fourier constants"  ''Proc. London Math. Soc. (2)'' , '''12'''  (1913)  pp. 71–88</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Hausdorff,  "Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen"  ''Math. Z.'' , '''16'''  (1923)  pp. 163–169</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. de Leeuw,  J.P. Kahane,  Y. Katznelson,  "Sur les coefficients de Fourier des fonctions continues"  ''C.R. Acad. Sci. Paris'' , '''285'''  (1977)  pp. 1001–1003</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.H. Young,  "On the determination of the summability of a function by means of its Fourier constants"  ''Proc. London Math. Soc. (2)'' , '''12'''  (1913)  pp. 71–88</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Hausdorff,  "Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen"  ''Math. Z.'' , '''16'''  (1923)  pp. 163–169</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. de Leeuw,  J.P. Kahane,  Y. Katznelson,  "Sur les coefficients de Fourier des fonctions continues"  ''C.R. Acad. Sci. Paris'' , '''285'''  (1977)  pp. 1001–1003</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Taking for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675028.png" /> the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675029.png" /> gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675030.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046750/h04675031.png" />.
+
Taking for $  g $
 +
the series $  \sum _ {1}  ^  \infty  a _ {n} \phi _ {n} $
 +
gives $  a _ {n} = c _ {n} ( g) $
 +
for all $  n \geq  1 $.

Latest revision as of 22:10, 5 June 2020


Estimates of the Fourier coefficients of functions in $ L _ {p} $; established by W.H. Young [1] and F. Hausdorff [2]. Let $ \phi _ {n} $ be an orthonormal system of functions on $ [ a, b] $, let $ | \phi _ {n} ( t) | \leq M $ for all $ t \in [ a, b] $ and for all $ n = 1, 2 \dots $ and let $ 1 < p \leq 2 $, $ 1/p + 1/p ^ \prime = 1 $. If $ f \in L _ {p} $, then

$$ \tag{1 } \left ( \sum _ {n = 1 } ^ \infty | c _ {n} ( f ) | ^ {p ^ \prime } \right ) ^ {1/p ^ \prime } \leq \ M ^ {( 2 - p) / p } \left ( \int\limits _ { a } ^ { b } | f ( t) | ^ {p} dt \right ) ^ {1/p} , $$

where $ c _ {n} ( f ) $ are the Fourier coefficients of $ f $. If $ \sum _ {n = 1 } ^ \infty | a _ {n} | ^ {p} $ converges, there exists a function $ g $ such that

$$ \tag{2 } \left ( \int\limits _ { a } ^ { b } | g ( t) | ^ {p ^ \prime } dt \right ) ^ {1/p ^ \prime } \leq \ M ^ {( 2 - p) / p } \left ( \sum _ {n = 1 } ^ \infty | a _ {n} | ^ {p} \right ) ^ {1/p} . $$

For $ g $ one may take $ \sum _ {n = 1 } ^ \infty a _ {n} \phi _ {n} $, and this series converges in $ L _ {p ^ \prime } $.

The Hausdorff–Young inequalities (1) and (2) are equivalent. For $ p > 2 $ they do not hold. Moreover, if $ b _ {n} \geq 0 $ and if $ \sum _ {n = 1 } ^ \infty b _ {n} ^ {2} < \infty $, then there exists a continuous function $ f $ such that its Fourier coefficients $ c _ {n} ( f ) $ in the trigonometric system satisfy the condition $ | c _ {n} ( f ) | > b _ {n} $. A qualitative statement of the Hausdorff–Young inequality (if $ f \in L _ {p} $, $ 1 \leq p \leq 2 $, then $ \{ c _ {n} ( f ) \} \in l _ {p ^ \prime } $) for unbounded orthonormal systems of functions does not hold, in general. An analogue of the Hausdorff–Young inequalities is valid for a broad class of function spaces.

References

[1] W.H. Young, "On the determination of the summability of a function by means of its Fourier constants" Proc. London Math. Soc. (2) , 12 (1913) pp. 71–88
[2] F. Hausdorff, "Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen" Math. Z. , 16 (1923) pp. 163–169
[3] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[4] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[5] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[6] K. de Leeuw, J.P. Kahane, Y. Katznelson, "Sur les coefficients de Fourier des fonctions continues" C.R. Acad. Sci. Paris , 285 (1977) pp. 1001–1003
[7] S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)

Comments

Taking for $ g $ the series $ \sum _ {1} ^ \infty a _ {n} \phi _ {n} $ gives $ a _ {n} = c _ {n} ( g) $ for all $ n \geq 1 $.

How to Cite This Entry:
Hausdorff-Young inequalities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff-Young_inequalities&oldid=47198
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article