View source for Lebesgue inequality

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>

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.J. Rivlin,   "An introduction to the approximation of functions" , Blaisdell  (1969)  pp. Sect. 4.1</TD></TR></table>