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Convergence of Fourier series - Summability | | Convergence of Fourier series - Summability: Encyclopedia II - Convergence of Fourier series - Summability | | Does the sequence 0,1,0,1,0,1,... converge to ½? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any sequence an is Cesàro summable to some a if
It is not difficult to see that if a sequence converges to some a then it is also Cesàro summable to it.
To discuss summability of Fourier series, we must replace SN wit ...
See also:Convergence of Fourier series, Convergence of Fourier series - Preliminaries, Convergence of Fourier series - Convergence at a given point., Convergence of Fourier series - Norm convergence, Convergence of Fourier series - Convergence almost everywhere, Convergence of Fourier series - Absolute convergence, Convergence of Fourier series - Summability, Convergence of Fourier series - Order of growth, Convergence of Fourier series - Multiple dimensions | | | Convergence of Fourier series, Convergence of Fourier series - Absolute convergence, Convergence of Fourier series - Convergence almost everywhere, Convergence of Fourier series - Convergence at a given point., Convergence of Fourier series - Multiple dimensions, Convergence of Fourier series - Norm convergence, Convergence of Fourier series - Order of growth, Convergence of Fourier series - Preliminaries, Convergence of Fourier series - Summability | | |
| | | Convergence of Fourier series: Encyclopedia II - Convergence of Fourier series - Summability
Convergence of Fourier series - Summability
Does the sequence 0,1,0,1,0,1,... converge to ½? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any sequence an is Cesàro summable to some a if
It is not difficult to see that if a sequence converges to some a then it is also Cesàro summable to it.
To discuss summability of Fourier series, we must replace SN with an appropriate notion. Hence we define
and ask: does KN(f) converge to f? KN is no longer associated with Dirichlet's kernel, but with Fejér's kernel, namely
where FN is Fejér's kernel,
The main difference is that Fejér's kernel is a positive kernel. This implies much better convergence properties
- If f is continuous at t then the Fourier series of f is summable at t to f(t). If f is continuous, its Fourier series is uniformly summable (i.e. KN converges uniformly to f).
- For any integrable f, KN converges to f in the L1 norm.
- There is no Gibbs phenomenon.
Results about summability can also imply results about regular convergence. For example, we learn that if f is continuous at t, then the Fourier series of f cannot converge to a value different from f(t). It may either converge to f(t) or diverge. This is because, if SN(f;t) converges to some value x, it is also summable to it, so from the first summability property above, x = f(t).
Other related archives1920s, 1966, L2, Lp spaces, Lp, Banach algebra, Big O notation, Cesàro mean, Cesàro summable, Dini test, Dirichlet's kernel, Fejér's kernel, Fourier series, Gibbs phenomenon, Hardy spaces, Hunt, Katznelson, Kolmogorov, Lennart Carleson, Nikolai Lusin, Norbert Wiener, Riesz-Fischer theorem, Riesz-Thorin interpolation theorem, Salomon Bochner, absolute convergence, absolutely converging, almost everywhere, bounded variation, converges, convolution, differentiable, first Baire category, function, harmonic sum, holomorphic functions, if and only if, integrable, mathematics, measure, periodic function, pointwise convergence, pure mathematics, square-integrable, summability methods, uniform boundedness principle, uniform convergence
 Adapted from the Wikipedia article "Summability", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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