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Difference between revisions of "De la Vallée-Poussin singular integral"

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An integral of the form
 
An integral of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302601.png" /></td> </tr></table>
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$$V_n(f;x)=\frac{1}{2\pi}\frac{(2n)!!}{(2n-1)!!}\int\limits_{-\pi}^\pi f(x+t)\cos^{2n}\frac t2dt$$
  
(see also [[De la Vallée-Poussin summation method|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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(see also [[De la Vallée-Poussin summation method|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 $(-\infty,\infty)$ [[#References|[1]]]. If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d0302607.png" /></td> </tr></table>
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$$\left(\int\limits_{-\pi}^xf(t)dt\right)_x'=f(x)$$
  
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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at a point $x$, then $V_n(f;x)\to f(x)$ as $n\to\infty$. The following equality is valid [[#References|[2]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d03026011.png" /></td> </tr></table>
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$$V_n(f;x)-f(x)=\frac{f''(x)}{n}+o\left(\frac1n\right).$$
  
 
====References====
 
====References====
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====Comments====
 
====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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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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030260/d03026017.png" /></td> </tr></table>
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$$\frac{(2n)!!}{(2n-1)!!}=\frac{2^{2n}(n!)^2}{(2n)!}.$$

Latest revision as of 16:26, 27 September 2014

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

$$\left(\int\limits_{-\pi}^xf(t)dt\right)_x'=f(x)$$

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

$$V_n(f;x)-f(x)=\frac{f''(x)}{n}+o\left(\frac1n\right).$$

References

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


Comments

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,

$$\frac{(2n)!!}{(2n-1)!!}=\frac{2^{2n}(n!)^2}{(2n)!}.$$

How to Cite This Entry:
De la Vallée-Poussin singular integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_singular_integral&oldid=33425
This article was adapted from an original article by P.P. Korovkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article