View source for Hausdorff-Young inequalities

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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

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

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