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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|trigonometric series]]. Let a series
 
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|trigonometric series]]. Let a 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/r/r081/r081880/r0818801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
  
with bounded sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r0818802.png" /> be given. The Riemann function for this series is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r0818803.png" /> obtained by twice term-by-term integration of the series:
+
\frac{a _ {0} }{2}
 +
+ \sum _ { n= } 1 ^  \infty  ( a _ {n}  \cos  nx + b _ {n}  \sin  nx )
 +
$$
  
<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/r/r081/r081880/r0818804.png" /></td> </tr></table>
+
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:
  
<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/r/r081/r081880/r0818805.png" /></td> </tr></table>
+
$$
 +
F( x)  =
 +
\frac{a _ {0} x  ^ {2} }{4}
 +
- \sum _ { n= } 1 ^  \infty 
 +
\frac{1}{n  ^ {2} }
 +
( a _ {n}  \cos  nx + b _ {n}  \sin  nx ) + Cx + D,
 +
$$
  
Riemann's first theorem: Let the series (*) converge at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r0818806.png" /> to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r0818807.png" />. The Schwarzian derivative (cf. [[Riemann derivative|Riemann derivative]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r0818808.png" /> then equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r0818809.png" />. Riemann's second theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188010.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188011.png" />. Then at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188012.png" />,
+
$$
 +
C, D  = \textrm{ const } .
 +
$$
  
<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/r/r081/r081880/r08188013.png" /></td> </tr></table>
+
Riemann's first theorem: Let the series (*) converge at a point  $  x _ {0} $
 +
to a number  $  S $.  
 +
The Schwarzian derivative (cf. [[Riemann derivative|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 $,
  
moreover, the convergence is uniform on any interval, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188014.png" /> is a uniformly smooth function.
+
$$
 +
\lim\limits _ {n \rightarrow \infty } 
 +
\frac{F( x+ h) + F( x- h) - 2F( x) }{h}
 +
  = 0;
 +
$$
  
If the series (*) converges on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188015.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188016.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188018.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188019.png" /> and
+
moreover, the convergence is uniform on any interval, that is, $  F $
 +
is a uniformly smooth function.
  
<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/r/r081/r081880/r08188020.png" /></td> </tr></table>
+
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
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188021.png" /> be real numbers tending to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188022.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188023.png" />, let
+
$$
 +
F( x)  = \int\limits _ { 0 } ^ { x }  dt \int\limits _ { 0 } ^ { t }  f( u)  du + Cx + D.
 +
$$
  
<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/r/r081/r081880/r08188024.png" /></td> </tr></table>
+
Let  $  a _ {n} , b _ {n} $
 +
be real numbers tending to  $  0 $
 +
as  $  n \rightarrow \infty $,
 +
let
  
be finite at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188025.png" />, and let
+
$$
 +
\underline{S} ( x)  = \
 +
\lim\limits _ {\overline{ {n \rightarrow \infty }}\; }  S _ {n} ( x) \  \textrm{ and } \  \overline{S}\; ( x)  = \overline{\lim\limits}\;
 +
_ {n \rightarrow \infty }  S _ {n} ( x)
 +
$$
  
<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/r/r081/r081880/r08188026.png" /></td> </tr></table>
+
be finite at a point  $  x $,
 +
and let
  
Then the upper and lower Schwarzian derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188028.png" /> belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081880/r08188030.png" /> is an absolute constant. (The du Bois-Reymond lemma.)
+
$$
 +
S( x)  = 
 +
\frac{1}{2}
 +
( \underline{S} ( x) + \overline{S}\; ( x)),\ \
 +
\delta ( x)  = 
 +
\frac{1}{2}
 +
( \overline{S}\; ( x) - \underline{S} ( x)).
 +
$$
 +
 
 +
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====
 
====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>
 
<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====
 
====Comments====

Revision as of 08:11, 6 June 2020


The Riemann function in the theory of trigonometric series is a function introduced by B. Riemann (1851) (see [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

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

Comments

See also Riemann summation method.

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

How to Cite This Entry:
Riemann function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_function&oldid=14654
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article