De la Vallée-Poussin singular integral

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An integral of the form

(see also de la Vallée-Poussin summation method). The sequence converges uniformly to for functions which are continuous and -periodic on [1]. If

at a point , then as . 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 stands for ( terms), and (also 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