Difference between revisions of "Lebesgue function"
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| + | $#C+1 = 10 : ~/encyclopedia/old_files/data/L057/L.0507840 Lebesgue function | ||
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A function | A function | ||
| − | + | $$ | |
| + | L _ {n} ^ \Phi ( t) = \int\limits _ { a } ^ { b } | ||
| + | \left | \sum _ { k= } 1 ^ { n } \phi _ {k} ( x) \phi _ {k} ( t) \right | d x ,\ t \in [ a , b ] , | ||
| + | $$ | ||
| − | where | + | where $ \Phi = \{ \phi _ {k} \} _ {k=} 1 ^ \infty $ |
| + | is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval $ [ a , b ] $, | ||
| + | $ n = 1 , 2 , . . . $. | ||
| + | Lebesgue functions are defined similarly in the case when an orthonormal system is specified on an arbitrary measure space. One has | ||
| − | + | $$ | |
| + | L _ {n} ^ \Phi ( t) = \ | ||
| + | \sup _ {f : \| f \| _ {C [ a , b ] } | ||
| + | \leq 1 } | S _ {n} ( f ) | ,\ \ | ||
| + | t \in [ a , b ] , | ||
| + | $$ | ||
where | where | ||
| − | + | $$ | |
| + | S _ {n} ( f ) ( t) = \sum _ { k= } 1 ^ { n } | ||
| + | c _ {k} ( f ) \phi _ {k} ( t) | ||
| + | $$ | ||
| − | is the | + | is the $ n $- |
| + | th partial sum of the [[Fourier series|Fourier series]] of $ f $ | ||
| + | with respect to $ \Phi $. | ||
| + | In the case when $ \Phi $ | ||
| + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)</TD></TR></table> | ||
Revision as of 22:16, 5 June 2020
A function
$$ L _ {n} ^ \Phi ( t) = \int\limits _ { a } ^ { b } \left | \sum _ { k= } 1 ^ { n } \phi _ {k} ( x) \phi _ {k} ( t) \right | d x ,\ t \in [ a , b ] , $$
where $ \Phi = \{ \phi _ {k} \} _ {k=} 1 ^ \infty $ is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval $ [ a , b ] $, $ n = 1 , 2 , . . . $. Lebesgue functions are defined similarly in the case when an orthonormal system is specified on an arbitrary measure space. One has
$$ L _ {n} ^ \Phi ( t) = \ \sup _ {f : \| f \| _ {C [ a , b ] } \leq 1 } | S _ {n} ( f ) | ,\ \ t \in [ a , b ] , $$
where
$$ S _ {n} ( f ) ( t) = \sum _ { k= } 1 ^ { n } c _ {k} ( f ) \phi _ {k} ( t) $$
is the $ n $- th partial sum of the Fourier series of $ f $ with respect to $ \Phi $. In the case when $ \Phi $ is the trigonometric system, the Lebesgue functions are constant and reduce to the Lebesgue constants. They were introduced by H. Lebesgue.
References
| [1] | S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951) |
How to Cite This Entry:
Lebesgue function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_function&oldid=13866
Lebesgue function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_function&oldid=13866
This article was adapted from an original article by B.S. Kashin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article