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An [[Infinite product|infinite product]] of the form
 
An [[Infinite product|infinite product]] of the form
  
<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/r082/r082280/r0822801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
 +
\prod _ { k= } 1 ^  \infty  ( 1 + \alpha _ {k}  \cos  n _ {k} x),\ \
 +
x \in [ 0, \pi ],
 +
$$
 +
 
 +
$$
  
<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/r082/r082280/r0822802.png" /></td> </tr></table>
+
\frac{n _ {k+} 1 }{n _ {k} }
 +
  \geq  q  > 1,\  | a _ {k} |  \leq  1,\ \
 +
\forall k \in \mathbf N .
 +
$$
  
With the help of such products (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r0822803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r0822804.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r0822805.png" />) F. Riesz indicated the first example of a continuous function of bounded variation whose Fourier coefficients are not of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r0822806.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r0822807.png" />, then the identity
+
With the help of such products ( $  a _ {k} = 1 $,  
 +
$  n _ {k} = 3  ^ {k} $
 +
for all $  k \in \mathbf N $)  
 +
F. Riesz indicated the first example of a continuous function of bounded variation whose Fourier coefficients are not of order $  o( 1/n) $.  
 +
If $  q > 3 $,  
 +
then the identity
  
<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/r082/r082280/r0822808.png" /></td> </tr></table>
+
$$
 +
\prod _ { k= } 1 ^ { m }  ( 1 + a _ {k}  \cos  n _ {k} x)  = \
 +
1 + \sum _ { k= } 1 ^ { {p _ m} } \gamma _ {k} \cos  kx,
 +
$$
  
<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/r082/r082280/r0822809.png" /></td> </tr></table>
+
$$
 +
p _ {m}  = n _ {1} + \dots + n _ {m} ,\  m \in \mathbf N ,\  x \in [ 0, \pi ],
 +
$$
  
 
gives the series
 
gives 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/r082/r082280/r08228010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
1 + \sum _ { k= } 1 ^  \infty  \gamma _ {k}  \cos  kx ,
 +
$$
  
which is said to represent the Riesz product (1). In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228013.png" />, the series (2) is the Fourier–Stieltjes series of a non-decreasing continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228015.png" /> and
+
which is said to represent the Riesz product (1). In case $  q \geq  3 $,  
 +
$  - 1 \leq  a _ {k} \leq  1 $
 +
for all $  k \in \mathbf N $,  
 +
the series (2) is the Fourier–Stieltjes series of a non-decreasing continuous function $  F $.  
 +
If $  q > 3 $
 +
and
  
<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/r082/r082280/r08228016.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 1 ^  \infty  a _ {k}  ^ {2}  = + \infty ,\ \
 +
- 1 \leq  a _ {k} \leq  1 ,\  \forall k \in \mathbf N ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228017.png" /> almost-everywhere. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228018.png" />, then the series (2) converges to zero almost-everywhere.
+
then $  F ^ { \prime } ( x) = 0 $
 +
almost-everywhere. If, in addition, $  a _ {k} \rightarrow 0 $,  
 +
then the series (2) converges to zero almost-everywhere.
  
A number of problems, mainly in the theory of [[Trigonometric series|trigonometric series]], has been solved using a natural generalization of the Riesz product when in (1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228019.png" /> is replaced by specially chosen trigonometric polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082280/r08228020.png" />.
+
A number of problems, mainly in the theory of [[Trigonometric series|trigonometric series]], has been solved using a natural generalization of the Riesz product when in (1) $  a _ {k}  \cos  n _ {k} x $
 +
is replaced by specially chosen trigonometric polynomials $  T _ {k} ( x) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>

Revision as of 08:11, 6 June 2020


An infinite product of the form

$$ \tag{1 } \prod _ { k= } 1 ^ \infty ( 1 + \alpha _ {k} \cos n _ {k} x),\ \ x \in [ 0, \pi ], $$

$$ \frac{n _ {k+} 1 }{n _ {k} } \geq q > 1,\ | a _ {k} | \leq 1,\ \ \forall k \in \mathbf N . $$

With the help of such products ( $ a _ {k} = 1 $, $ n _ {k} = 3 ^ {k} $ for all $ k \in \mathbf N $) F. Riesz indicated the first example of a continuous function of bounded variation whose Fourier coefficients are not of order $ o( 1/n) $. If $ q > 3 $, then the identity

$$ \prod _ { k= } 1 ^ { m } ( 1 + a _ {k} \cos n _ {k} x) = \ 1 + \sum _ { k= } 1 ^ { {p _ m} } \gamma _ {k} \cos kx, $$

$$ p _ {m} = n _ {1} + \dots + n _ {m} ,\ m \in \mathbf N ,\ x \in [ 0, \pi ], $$

gives the series

$$ \tag{2 } 1 + \sum _ { k= } 1 ^ \infty \gamma _ {k} \cos kx , $$

which is said to represent the Riesz product (1). In case $ q \geq 3 $, $ - 1 \leq a _ {k} \leq 1 $ for all $ k \in \mathbf N $, the series (2) is the Fourier–Stieltjes series of a non-decreasing continuous function $ F $. If $ q > 3 $ and

$$ \sum _ { k= } 1 ^ \infty a _ {k} ^ {2} = + \infty ,\ \ - 1 \leq a _ {k} \leq 1 ,\ \forall k \in \mathbf N , $$

then $ F ^ { \prime } ( x) = 0 $ almost-everywhere. If, in addition, $ a _ {k} \rightarrow 0 $, then the series (2) converges to zero almost-everywhere.

A number of problems, mainly in the theory of trigonometric series, has been solved using a natural generalization of the Riesz product when in (1) $ a _ {k} \cos n _ {k} x $ is replaced by specially chosen trigonometric polynomials $ T _ {k} ( x) $.

References

[1] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Riesz product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_product&oldid=48567
This article was adapted from an original article by V.F. Emel'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article