De la Vallée-Poussin singular integral

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An integral of the form

$$V_n(f;x)=\frac{1}{2\pi}\frac{(2n)!!}{(2n-1)!!}\int\limits_{-\pi}^\pi f(x+t)\cos^{2n}\frac t2dt$$

(see also de la Vallée-Poussin summation method). The sequence $V_n(f;x)$ converges uniformly to $f(x)$ for functions $f$ which are continuous and $2\pi$-periodic on $(-\infty,\infty)$ [1]. If


at a point $x$, then $V_n(f;x)\to f(x)$ as $n\to\infty$. The following equality is valid [2]:



[1] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[2] I.P. Natanson, "Constructive function theory" , 1 , F. Ungar (1964) (Translated from Russian)


The notation $(2m)!!$ stands for $2m\cdot(2m-2)\cdots2$ ($m$ terms), and $(2m-1)!!=(2m-1)(2m-3)\cdots3\cdot1$ (also $m$ terms). Thus,


How to Cite This Entry:
De la Vallée-Poussin singular integral. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by P.P. Korovkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article