Difference between revisions of "Fejér sum"
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One of the arithmetic means of the partial sums of a Fourier series in the trigonometric system | One of the arithmetic means of the partial sums of a Fourier series in the trigonometric system | ||
− | + | $$ | |
+ | \sigma _ {n} ( f, x) = \ | ||
+ | { | ||
+ | \frac{1}{n + 1 } | ||
+ | } | ||
+ | \sum _ {k = 0 } ^ { n } | ||
+ | s _ {k} ( f, x) = | ||
+ | $$ | ||
− | + | $$ | |
+ | = \ | ||
+ | { | ||
+ | \frac{a _ {0} }{2} | ||
+ | } + \sum _ {k = 1 } ^ { n } \left ( 1 - { | ||
+ | \frac{k}{n + 1 } | ||
+ | } \right ) ( a _ {k} \cos kx + b _ {k} \sin kx), | ||
+ | $$ | ||
− | where | + | where $ a _ {k} $ |
+ | and $ b _ {k} $ | ||
+ | are the Fourier coefficients of the function $ f $. | ||
− | If | + | If $ f $ |
+ | is continuous, then $ \sigma _ {n} ( f, x) $ | ||
+ | converges uniformly to $ f ( x) $; | ||
+ | $ \sigma _ {n} ( f, x) $ | ||
+ | converges to $ f ( x) $ | ||
+ | in the metric of $ L $. | ||
− | If | + | If $ f $ |
+ | belongs to the class of functions that satisfy a Lipschitz condition of order $ \alpha < 1 $, | ||
+ | then | ||
− | + | $$ | |
+ | \| f ( x) - \sigma _ {n} ( f, x) \| _ {c} = \ | ||
+ | O \left ( { | ||
+ | \frac{1}{n ^ \alpha } | ||
+ | } \right ) , | ||
+ | $$ | ||
− | that is, in this case the Fejér sum approximates | + | that is, in this case the Fejér sum approximates $ f $ |
+ | at the rate of the best approximating functions of the indicated class. But Fejér sums cannot provide a high rate of approximation: The estimate | ||
− | + | $$ | |
+ | \| f ( x) - \sigma _ {n} ( f, x) \| _ {c} = \ | ||
+ | o \left ( { | ||
+ | \frac{1}{n} | ||
+ | } \right ) | ||
+ | $$ | ||
is valid only for constant functions. | is valid only for constant functions. | ||
Line 23: | Line 69: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Fejér, "Untersuchungen über Fouriersche Reihen" ''Math. Ann.'' , '''58''' (1903) pp. 51–69</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.P. Natanson, "Constructive function theory" , '''1–3''' , F. Ungar (1964–1965) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Fejér, "Untersuchungen über Fouriersche Reihen" ''Math. Ann.'' , '''58''' (1903) pp. 51–69</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.P. Natanson, "Constructive function theory" , '''1–3''' , F. Ungar (1964–1965) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
See also [[Fejér summation method|Fejér summation method]]. | See also [[Fejér summation method|Fejér summation method]]. |
Latest revision as of 19:38, 5 June 2020
One of the arithmetic means of the partial sums of a Fourier series in the trigonometric system
$$ \sigma _ {n} ( f, x) = \ { \frac{1}{n + 1 } } \sum _ {k = 0 } ^ { n } s _ {k} ( f, x) = $$
$$ = \ { \frac{a _ {0} }{2} } + \sum _ {k = 1 } ^ { n } \left ( 1 - { \frac{k}{n + 1 } } \right ) ( a _ {k} \cos kx + b _ {k} \sin kx), $$
where $ a _ {k} $ and $ b _ {k} $ are the Fourier coefficients of the function $ f $.
If $ f $ is continuous, then $ \sigma _ {n} ( f, x) $ converges uniformly to $ f ( x) $; $ \sigma _ {n} ( f, x) $ converges to $ f ( x) $ in the metric of $ L $.
If $ f $ belongs to the class of functions that satisfy a Lipschitz condition of order $ \alpha < 1 $, then
$$ \| f ( x) - \sigma _ {n} ( f, x) \| _ {c} = \ O \left ( { \frac{1}{n ^ \alpha } } \right ) , $$
that is, in this case the Fejér sum approximates $ f $ at the rate of the best approximating functions of the indicated class. But Fejér sums cannot provide a high rate of approximation: The estimate
$$ \| f ( x) - \sigma _ {n} ( f, x) \| _ {c} = \ o \left ( { \frac{1}{n} } \right ) $$
is valid only for constant functions.
Fejér sums were introduced by L. Fejér [1].
References
[1] | L. Fejér, "Untersuchungen über Fouriersche Reihen" Math. Ann. , 58 (1903) pp. 51–69 |
[2] | N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) |
[3] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |
[4] | I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) |
[5] | V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian) |
Comments
See also Fejér summation method.
Fejér sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fej%C3%A9r_sum&oldid=15688