View source for Lebesgue function

A function

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578402.png" /> is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578404.png" />. Lebesgue functions are defined similarly in the case when an orthonormal system is specified on an arbitrary measure space. One has

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where

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is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578407.png" />-th partial sum of the [[Fourier series|Fourier series]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578408.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578409.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l05784010.png" /> is the [[Trigonometric system|trigonometric system]], the Lebesgue functions are constant and reduce to the [[Lebesgue constants|Lebesgue constants]]. They were introduced by H. Lebesgue.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,   H. Steinhaus,   "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>