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The Riemann function in the theory of trigonometric series is a function introduced by B. Riemann (1851) (see [[#References|[1]]]) for studying the problem of the representation of a function by a [[trigonometric series]]. Let a series

$$ \tag{* }

\frac{a _ {0} }{2}
 + \sum _ { n= 1} ^  \infty   ( a _ {n}  \cos  nx + b _ {n}  \sin  nx )
$$

with bounded sequences  $  \{ a _ {n} \} , \{ b _ {n} \} $
be given. The Riemann function for this series is the function  $  F $
obtained by twice term-by-term integration of the series:

$$ 
F( x)  =  
\frac{a _ {0} x  ^ {2} }{4}
 - \sum _ { n= 1 }^  \infty   
\frac{1}{n  ^ {2} }
 ( a _ {n}  \cos  nx + b _ {n}  \sin  nx ) + Cx + D,
$$

$$ 
C, D  =  \textrm{ const } .
$$

Riemann's first theorem: Let the series (*) converge at a point  $  x _ {0} $
to a number  $  S $. 
The Schwarzian derivative (cf. [[Riemann derivative]])  $  D _ {2} F( x _ {0} ) $
then equals  $  S $. 
Riemann's second theorem: Let  $  a _ {n} , b _ {n} \rightarrow 0 $
as  $  n \rightarrow \infty $. 
Then at any point  $  x $,

$$ 
\lim\limits _ {n \rightarrow \infty }  
\frac{F( x+ h) + F( x- h) - 2F( x) }{h}
  =  0;
$$

moreover, the convergence is uniform on any interval, that is,  $  F $
is a uniformly smooth function.

If the series (*) converges on  $  [ 0, 2 \pi ] $
to  $  f( x) $
and if  $  f \in L[ 0, 2 \pi ] $, 
then  $  D _ {2} F( x) = f( x) $
for each  $  x \in [ 0, 2 \pi ] $
and

$$ 
F( x)  =  \int\limits _ { 0 } ^ { x }   dt \int\limits _ { 0 } ^ { t }  f( u)  du + Cx + D.
$$

Let  $  a _ {n} , b _ {n} $
be real numbers tending to  $  0 $
as  $  n \rightarrow \infty $, 
let

$$ 
\underline{S} ( x)  = \ 
\lim\limits _ {\overline{ {n \rightarrow \infty }}\; }  S _ {n} ( x) \  \textrm{ and } \  \overline{S}\; ( x)  =  \overline{\lim\limits}\;
_ {n \rightarrow \infty }  S _ {n} ( x)
$$

be finite at a point  $  x $, 
and let

$$ 
S( x)  =  
\frac{1}{2}
 ( \underline{S} ( x) + \overline{S}\; ( x)),\ \ 
\delta ( x)  =  
\frac{1}{2}
 ( \overline{S}\; ( x) - \underline{S} ( x)).
$$

Then the upper and lower Schwarzian derivatives  $  \overline{D}\; _ {2} F( x) $
and  $  \underline{D} _ {2} F( x) $
belong to  $  [ S - \mu \delta , S + \mu \delta ] $, 
where  $  \mu $
is an absolute constant. (The du Bois-Reymond lemma.)

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  B. Riemann,   "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , ''Gesammelte Math. Abhandlungen'' , Dover, reprint  (1957)  pp. 227–264</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR>
</table>

====Comments====
See also [[Riemann summation method|Riemann summation method]].

For Riemann's function in the theory of differential equations see [[Riemann method|Riemann method]].