Namespaces
Variants
Actions

Difference between revisions of "Riesz product"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (latex details)
 
Line 14: Line 14:
  
 
$$ \tag{1 }
 
$$ \tag{1 }
\prod _ { k= } 1 ^  \infty  ( 1 + \alpha _ {k}  \cos  n _ {k} x),\ \  
+
\prod_{k=1} ^  \infty  ( 1 + \alpha _ {k}  \cos  n _ {k} x),\ \  
 
x \in [ 0, \pi ],
 
x \in [ 0, \pi ],
 
$$
 
$$
Line 33: Line 33:
  
 
$$  
 
$$  
\prod _ { k= } 1 ^ { m }  ( 1 + a _ {k}  \cos  n _ {k} x)  = \  
+
\prod_{k=1} ^ { m }  ( 1 + a _ {k}  \cos  n _ {k} x)  = \  
1 + \sum _ { k= } 1 ^ { {p _ m} } \gamma _ {k} \cos  kx,
+
1 + \sum_{k=1} ^ { {p _ m} } \gamma _ {k} \cos  kx,
 
$$
 
$$
  
Line 44: Line 44:
  
 
$$ \tag{2 }
 
$$ \tag{2 }
1 + \sum _ { k= } 1 ^  \infty  \gamma _ {k}  \cos  kx ,
+
1 + \sum_{k=1}^  \infty  \gamma _ {k}  \cos  kx ,
 
$$
 
$$
  
Line 55: Line 55:
  
 
$$  
 
$$  
\sum _ { k= } 1 ^  \infty  a _ {k}  ^ {2}  =  + \infty ,\ \  
+
\sum _ {k=1} ^  \infty  a _ {k}  ^ {2}  =  + \infty ,\ \  
 
- 1 \leq  a _ {k} \leq  1 ,\  \forall k \in \mathbf N ,
 
- 1 \leq  a _ {k} \leq  1 ,\  \forall k \in \mathbf N ,
 
$$
 
$$

Latest revision as of 12:41, 6 January 2024


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=54871
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