Riesz product

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An infinite product of the form


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


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 .


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098