# Riesz product

An infinite product of the form

(1) |

With the help of such products (, for all ) F. Riesz indicated the first example of a continuous function of bounded variation whose Fourier coefficients are not of order . If , then the identity

gives the series

(2) |

which is said to represent the Riesz product (1). In case , for all , the series (2) is the Fourier–Stieltjes series of a non-decreasing continuous function . If and

then almost-everywhere. If, in addition, , 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) is replaced by specially chosen trigonometric polynomials .

#### 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. V.F. Emel'yanov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Riesz_product&oldid=16157