View source for De la Vallée-Poussin singular integral

An integral of the form

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(see also [[De la Vallée-Poussin summation method|de la Vallée-Poussin summation method]]). The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302602.png" /> converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302603.png" /> for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302604.png" /> which are continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302605.png" />-periodic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302606.png" /> [[#References|[1]]]. If

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at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302608.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302609.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d03026010.png" />. The following equality is valid [[#References|[2]]]:

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,   "Constructive function theory" , '''1''' , F. Ungar  (1964)  (Translated from Russian)</TD></TR></table>



====Comments====
The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d03026012.png" /> stands for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d03026013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d03026014.png" /> terms), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d03026015.png" /> (also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d03026016.png" /> terms). Thus,

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