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A summation method of arithmetical averages (cf. [[Arithmetical averages, summation method of|Arithmetical averages, summation method of]]), applied to the summation of Fourier series. It was first applied by L. Fejér [[#References|[1]]].
 
A summation method of arithmetical averages (cf. [[Arithmetical averages, summation method of|Arithmetical averages, summation method of]]), applied to the summation of Fourier series. It was first applied by L. Fejér [[#References|[1]]].
  
 
The Fourier series
 
The Fourier series
  
<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/f/f038/f038370/f0383701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\frac12+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)\label{1}\tag{1}$$
  
of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f0383702.png" /> is summable by the Fejér summation method to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f0383703.png" /> if
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of a function $f\in L(-\pi,\pi)$ is summable by the Fejér summation method to a function $s$ 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/f/f038/f038370/f0383704.png" /></td> </tr></table>
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$$\lim_{n\to\infty}\sigma_n(x)=s(x),$$
  
 
where
 
where
  
<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/f/f038/f038370/f0383705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$\sigma_n(x)=\frac{1}{n+1}\sum_{k=0}^ns_k(x),\label{2}\tag{2}$$
  
and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f0383706.png" /> are the partial sums of (1).
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and the $s_k(x)$ are the partial sums of \eqref{1}.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f0383707.png" /> is a point of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f0383708.png" /> or a discontinuity of the first kind, then its Fourier series at that point is Fejér summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f0383709.png" /> or to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837010.png" />, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837011.png" /> is continuous on some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837012.png" />, then its Fourier series is uniformly Fejér summable on every segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837013.png" />; and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837014.png" /> is continuous everywhere, then the series is summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837015.png" /> uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837016.png" /> (Fejér's theorem).
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If $x$ is a point of continuity of $f$ or a discontinuity of the first kind, then its Fourier series at that point is Fejér summable to $f(x)$ or to $(f(x+0)+f(x-0))/2$, respectively. If $f$ is continuous on some interval $(a,b)$, then its Fourier series is uniformly Fejér summable on every segment $[\alpha,\beta]\subset(a,b)$; and if $f$ is continuous everywhere, then the series is summable to $f$ uniformly on $[-\pi,\pi]$ (Fejér's theorem).
  
This result was strengthened by H. Lebesgue [[#References|[2]]], who proved that for every summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837017.png" />, its Fourier series is almost-everywhere Fejér summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837018.png" />.
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This result was strengthened by H. Lebesgue [[#References|[2]]], who proved that for every summable function $f$, its Fourier series is almost-everywhere Fejér summable to $f$.
  
 
The function
 
The function
  
<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/f/f038/f038370/f03837019.png" /></td> </tr></table>
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$$K_n(x)=\frac{1}{n+1}\sum_{k=0}^n\left(\frac12+\sum_{\nu=1}^k\cos\nu x\right)\equiv$$
  
<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/f/f038/f038370/f03837020.png" /></td> </tr></table>
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$$\equiv\frac{1}{2(n+1)}\left(\frac{\sin(n+1)x/2}{\sin x/2}\right)^2$$
  
is called the Fejér kernel. It can be used to express the Fejér means (2) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038370/f03837021.png" /> in the form
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is called the Fejér kernel. It can be used to express the Fejér means \eqref{2} of $f$ in 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/f/f038/f038370/f03837022.png" /></td> </tr></table>
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$$\sigma_n(x)=\frac1\pi\int_{-\pi}^\pi f(x+u)K_n(u)du.$$
  
 
====References====
 
====References====

Latest revision as of 17:25, 14 February 2020

A summation method of arithmetical averages (cf. Arithmetical averages, summation method of), applied to the summation of Fourier series. It was first applied by L. Fejér [1].

The Fourier series

$$\frac12+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)\label{1}\tag{1}$$

of a function $f\in L(-\pi,\pi)$ is summable by the Fejér summation method to a function $s$ if

$$\lim_{n\to\infty}\sigma_n(x)=s(x),$$

where

$$\sigma_n(x)=\frac{1}{n+1}\sum_{k=0}^ns_k(x),\label{2}\tag{2}$$

and the $s_k(x)$ are the partial sums of \eqref{1}.

If $x$ is a point of continuity of $f$ or a discontinuity of the first kind, then its Fourier series at that point is Fejér summable to $f(x)$ or to $(f(x+0)+f(x-0))/2$, respectively. If $f$ is continuous on some interval $(a,b)$, then its Fourier series is uniformly Fejér summable on every segment $[\alpha,\beta]\subset(a,b)$; and if $f$ is continuous everywhere, then the series is summable to $f$ uniformly on $[-\pi,\pi]$ (Fejér's theorem).

This result was strengthened by H. Lebesgue [2], who proved that for every summable function $f$, its Fourier series is almost-everywhere Fejér summable to $f$.

The function

$$K_n(x)=\frac{1}{n+1}\sum_{k=0}^n\left(\frac12+\sum_{\nu=1}^k\cos\nu x\right)\equiv$$

$$\equiv\frac{1}{2(n+1)}\left(\frac{\sin(n+1)x/2}{\sin x/2}\right)^2$$

is called the Fejér kernel. It can be used to express the Fejér means \eqref{2} of $f$ in the form

$$\sigma_n(x)=\frac1\pi\int_{-\pi}^\pi f(x+u)K_n(u)du.$$

References

[1] L. Fejér, "Untersuchungen über Fouriersche Reihen" Math. Ann. , 58 (1903) pp. 51–69
[2] H. Lebesgue, "Recherches sur la convergence de séries de Fourier" Math. Ann. , 61 (1905) pp. 251–280
[3] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[4] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)


Comments

See also Cesàro summation methods.

How to Cite This Entry:
Fejér summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fej%C3%A9r_summation_method&oldid=23272
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article