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Difference between revisions of "Lebesgue inequality"

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An estimate of the deviation of the partial sums of a [[Fourier series|Fourier series]] using best approximations. In the case of the trigonometric system, the Lebesgue inequality is understood to be the relation
 
An estimate of the deviation of the partial sums of a [[Fourier series|Fourier series]] using best approximations. In the case of the trigonometric system, the Lebesgue inequality is understood to be the relation
  
<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/l/l057/l057850/l0578501.png" /></td> </tr></table>
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$$\max_x|R_n(f,x)|\leq(L_n+1)E_n(f),\quad n=1,2,\ldots,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l0578502.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l0578503.png" />-th remainder of the (trigonometric) Fourier series of a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l0578504.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l0578505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l0578506.png" /> are the [[Lebesgue constants|Lebesgue constants]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l0578507.png" /> is the best uniform approximation (error) (cf. [[Best approximation|Best approximation]]) by trigonometric polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l0578508.png" />. The Lebesgue inequality is a relation of general character: analogues of it hold for arbitrary orthonormal systems with suitable definitions of the Lebesgue constants and of the best approximation, and also for the comparison of the remainders of Fourier series with best approximations in norms of other spaces, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l0578509.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057850/l05785010.png" />. The Lebesgue inequality and relations similar to it are often used in approximation theory to obtain estimates of best approximations from below. The inequality was established by H. Lebesgue.
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where $R_n(f,x)$ is the $n$-th remainder of the (trigonometric) Fourier series of a continuous $2\pi$-periodic function $f$, $L_n$ are the [[Lebesgue constants|Lebesgue constants]], and $E_n(f)$ is the best uniform approximation (error) (cf. [[Best approximation|Best approximation]]) by trigonometric polynomials of degree $n$. The Lebesgue inequality is a relation of general character: analogues of it hold for arbitrary orthonormal systems with suitable definitions of the Lebesgue constants and of the best approximation, and also for the comparison of the remainders of Fourier series with best approximations in norms of other spaces, for example, $L_p$, $1\leq p<\infty$. The Lebesgue inequality and relations similar to it are often used in approximation theory to obtain estimates of best approximations from below. The inequality was established by H. Lebesgue.
  
 
====References====
 
====References====

Latest revision as of 09:15, 1 August 2014

An estimate of the deviation of the partial sums of a Fourier series using best approximations. In the case of the trigonometric system, the Lebesgue inequality is understood to be the relation

$$\max_x|R_n(f,x)|\leq(L_n+1)E_n(f),\quad n=1,2,\ldots,$$

where $R_n(f,x)$ is the $n$-th remainder of the (trigonometric) Fourier series of a continuous $2\pi$-periodic function $f$, $L_n$ are the Lebesgue constants, and $E_n(f)$ is the best uniform approximation (error) (cf. Best approximation) by trigonometric polynomials of degree $n$. The Lebesgue inequality is a relation of general character: analogues of it hold for arbitrary orthonormal systems with suitable definitions of the Lebesgue constants and of the best approximation, and also for the comparison of the remainders of Fourier series with best approximations in norms of other spaces, for example, $L_p$, $1\leq p<\infty$. The Lebesgue inequality and relations similar to it are often used in approximation theory to obtain estimates of best approximations from below. The inequality was established by H. Lebesgue.

References

[1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)


Comments

References

[a1] T.J. Rivlin, "An introduction to the approximation of functions" , Blaisdell (1969) pp. Sect. 4.1
How to Cite This Entry:
Lebesgue inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_inequality&oldid=15437
This article was adapted from an original article by K.I. Oskolkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article