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Let  $  \{ \phi _ {n} \} $
be an [[Orthonormal system|orthonormal system]] of functions on an interval  $  [ a, b] $
and let  $  | \phi _ {n} | \leq  M $
almost everywhere on  $  [ a, b] $
for any  $  n $.

a) If  $  f \in L _ {p} [ a, b] $, 
$  1 < p \leq  2 $, 
then its Fourier coefficients with respect to  $  \{ \phi _ {n} \} $,

$$ 
c _ {n}  =  \int\limits _ { a } ^ { b }  f \overline \phi \; _ {n}  dx
$$

satisfy the Riesz inequality

$$ 
\| \{ c _ {n} \} \| _ {q}  \leq   M  ^ {2/p-} 1 \| f \| _ {p} ,\ \ 

\frac{1}{p}
 + 
\frac{1}{q}
 = 1.
$$

b) For any sequence  $  \{ c _ {n} \} $
with  $  \| \{ c _ {n} \} \| _ {p} < \infty $, 
$  1 < p \leq  2 $, 
there exists a function  $  f \in L _ {q} [ a, b] $
with  $  c _ {n} $
as its Fourier coefficients and satisfying the Riesz inequality

$$ 
\| f \| _ {q}  \leq   M  ^ {2/p-} 1 \| \{ c _ {n} \} \| _ {p} ,\ \ 

\frac{1}{p}
 + 
\frac{1}{q}
 = 1.
$$

If  $  f \in L _ {p} [ 0, 2 \pi ] $, 
$  1 < p \leq  \infty $, 
then the [[Conjugate function|conjugate function]]  $  \overline{f}\; \in L _ {p} [ 0, 2 \pi ] $
and the Riesz inequality

$$ 
\| \overline{f}\; \| _ {p}  \leq   A _ {p} \| f \| _ {p}  $$

holds, where  $  A _ {p} $
is a constant depending only on  $  p $.

Assertion 1) was for the first time proved by F. Riesz [[#References|[1]]]; particular cases of it were studied earlier by W.H. Young and F. Hausdorff. Assertion 2) was first proved by M. Riesz [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Riesz,   "Ueber eine Verallgemeinerung der Parsevalschen Formel"  ''Math. Z.'' , '''18'''  (1923)  pp. 117–124</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Riesz,   "Sur les fonctions conjuguées"  ''Math. Z.'' , '''27'''  (1927)  pp. 218–244</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">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>

====Comments====
For 2) see also [[Interpolation of operators|Interpolation of operators]] (it is a consequence of the Marcinkiewicz interpolation theorem and the weak type  $  ( 1, 1) $
of the conjugation operator) and [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Butzer,   R.J. Nessel,   "Fourier analysis and approximation" , '''1''' , Birkhäuser  (1971)  pp. Chapt. 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Hausdorff,   "Eine Ausdehnung des Parsevalschen Satzes über Fourier-reihen"  ''Math. Z.'' , '''16'''  (1923)  pp. 163–169</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.M. Stein,   G. Weiss,   "Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1975)  pp. Chapt. VI, §5</TD></TR></table>