Riesz product

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)