Difference between revisions of "De la Vallée-Poussin singular integral"
From Encyclopedia of Mathematics
Ulf Rehmann (talk | contribs) m (moved De la Vallee-Poussin singular integral to De la Vallée-Poussin singular integral over redirect: accented title) |
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An integral of the form | 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|de la Vallée-Poussin summation method]]). The sequence | + | (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 |
| − | + | $$\left(\int\limits_{-\pi}^xf(t)dt\right)_x'=f(x)$$ | |
| − | at a point | + | at a point $x$, then $V_n(f;x)\to f(x)$ as $n\to\infty$. The following equality is valid [[#References|[2]]]: |
| − | + | $$V_n(f;x)-f(x)=\frac{f''(x)}{n}+o\left(\frac1n\right).$$ | |
====References==== | ====References==== | ||
| Line 17: | Line 18: | ||
====Comments==== | ====Comments==== | ||
| − | The notation | + | 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, |
| − | + | $$\frac{(2n)!!}{(2n-1)!!}=\frac{2^{2n}(n!)^2}{(2n)!}.$$ | |
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=23248
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=23248
This article was adapted from an original article by P.P. Korovkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article