Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Convergence of Fourier series
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Absolute convergence== A function ''ƒ'' has an [[Absolute convergence|absolutely converging]] Fourier series if :<math>\|f\|_A:=\sum_{n=-\infty}^\infty |\widehat{f}(n)|<\infty.</math> If this condition holds then <math>(S_N f)(x)</math> converges absolutely for every <math>x</math>. Conversely, for this condition to hold, it suffices that <math>(S_N f)(x)</math> converges absolutely for some <math>x</math>. In other words, for absolute convergence there is no issue of ''where'' the sum converges absolutely — if it converges absolutely at one point then it does so everywhere. The family of all functions with absolutely converging Fourier series is a type of [[Banach algebra]] called the [[Wiener algebra]], after [[Norbert Wiener]], who proved that if ''ƒ'' has absolutely converging Fourier series and is never zero, then 1/''ƒ'' has absolutely converging Fourier series. A simplification of the original proof of Wiener's theorem was given by [[Israel Gelfand]] and later by [[Donald J. Newman]] in 1975. [[Sergei Bernstein]]'s theorem<ref>Teschl, Theorem 8.10</ref> states that, if <math>f</math> belongs to a α-Hölder class for α > 1/2 then{{citation needed|reason=The Teschl reference insufficiently covers the statement. The Krein algebra, for example, isn't discussed at all.|date=December 2024}} :<math>\|f\|_A\le c_\alpha \|f\|_{{\rm Lip}_\alpha},\qquad \|f\|_K:=\sum_{n=-\infty}^{+\infty} |n| |\widehat{f}(n)|^2\le c_\alpha \|f\|^2_{{\rm Lip}_\alpha}</math> for <math>\|f\|_{{\rm Lip}_\alpha}</math> the constant in the [[Hölder condition]], <math>c_\alpha</math> a constant only dependent on <math>\alpha</math>; <math>\|f\|_K</math> is the norm of the Krein algebra. Notice that the 1/2 here is essential—there is an example of a 1/2-Hölder functions due to Hardy and Littlewood,<ref>Teschl, Example 8.10</ref> which do not belong to the Wiener algebra. Besides, this theorem cannot improve the best known bound on the size of the Fourier coefficient of a α-Hölder function—that is only <math>O(1/n^\alpha)</math> and then not summable. Zygmund's theorem states that, if ''ƒ'' is of [[bounded variation]] ''and'' belongs to a α-Hölder class for some α > 0, it belongs to the Wiener algebra.<ref>Teschl, Theorem 8.11</ref> ==Norm convergence== According to the [[Riesz–Fischer theorem]], if ''ƒ'' is [[square-integrable]] then <math>S_N (f)</math> converges to ''ƒ'' in the [[Lp_space#Lp_spaces_and_Lebesgue_integrals|{{math|''L''<sup>2</sup>}}-norm]], that is <math display="block">\lim_{N\rightarrow\infty}\int_0^{2\pi}\left|f(x)-S_N(f) (x) \right|^2\,\mathrm{d}x=0.</math> The converse is also true: if the limit above is zero, then <math>f</math> must be in <math>\in L^2</math>. More generally, for <math>f\in L^p</math>, convergence in the {{math|''L''<sup>''p''</sup>}}-norm holds if <math>1< p< \infty</math>.<ref>Teschl, Theorem 8.4</ref> The original proof uses properties of [[holomorphic function]]s and [[Hardy space]]s, and another proof, due to [[Salomon Bochner]] relies upon the [[Riesz–Thorin theorem|Riesz–Thorin interpolation theorem]]. For ''p'' = 1 and infinity, the result is not true. The construction of an example of divergence in ''L''<sup>1</sup> was first done by [[Andrey Kolmogorov]] (see below). For infinity, the result is a corollary of the [[uniform boundedness principle]]. If the partial sum ''S<sub>N</sub>'' is replaced by a suitable [[summability kernel]] (for example the ''Fejér sum'' obtained by convolution with the [[Fejér kernel]]), basic functional analytic techniques can be applied to show that norm convergence holds for 1 ≤ ''p'' < ∞. ==Convergence almost everywhere== The problem whether the Fourier series of any continuous function converges [[almost everywhere]] was posed by [[Nikolai Lusin]] in the 1920s. It was resolved positively in 1966 by [[Lennart Carleson]]. His result, now known as [[Carleson's theorem]], tells the Fourier expansion of any function in ''L''<sup>2</sup> converges almost everywhere. Later on, [[Richard Hunt (mathematician)|Richard Hunt]] generalized this to ''L''<sup>''p''</sup> for any ''p'' > 1. Contrariwise, [[Andrey Kolmogorov]], in his first scientific work, constructed an example of a function in ''L''<sup>1</sup> whose Fourier series diverges almost everywhere (later improved to diverge everywhere). [[Jean-Pierre Kahane]] and [[Yitzhak Katznelson (mathematician)|Yitzhak Katznelson]] proved that for any given set ''E'' of [[measure (mathematics)|measure]] zero, there exists a continuous function ''ƒ'' such that the Fourier series of ''ƒ'' fails to converge on any point of ''E''. ==Summability== Does the sequence 0,1,0,1,0,1,... (the partial sums of [[Grandi's series]]) converge to {{sfrac|1|2}}? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any sequence <math>(a_n)_{n=1}^\infty</math> is [[Cesàro mean|Cesàro summable]] to some ''a'' if :<math>\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n s_k=a.</math> Where with <math>s_k</math> we denote the {{mvar|k}}th [[partial sum]]: :<math>s_k = a_1 + \cdots + a_k= \sum_{n=1}^k a_n</math> It is not difficult to see that if a sequence converges to some ''a'' then it is also [[Cesàro mean|Cesàro summable]] to it. To discuss summability of Fourier series, we must replace <math>S_N</math> with an appropriate notion. Hence we define :<math>K_N(f;t)=\frac{1}{N}\sum_{n=0}^{N-1} S_n(f;t), \quad N \ge 1,</math> and ask: does <math>K_N(f)</math> converge to ''f''? <math>K_N </math> is no longer associated with Dirichlet's kernel, but with [[Fejér kernel|Fejér's kernel]], namely :<math>K_N(f)=f*F_N\,</math> where <math>F_N</math> is Fejér's kernel, :<math>F_N=\frac{1}{N}\sum_{n=0}^{N-1} D_n.</math> The main difference is that Fejér's kernel is a positive kernel. [[Fejér's theorem]] states that the above sequence of partial sums converge uniformly to ''ƒ''. This implies much better convergence properties * If ''ƒ'' is continuous at ''t'' then the Fourier series of ''ƒ'' is summable at ''t'' to ''ƒ''(''t''). If ''ƒ'' is continuous, its Fourier series is uniformly summable (i.e. <math>K_N(f)</math> converges uniformly to ''ƒ''). * For any integrable ''ƒ'', <math>K_N(f)</math> converges to ''ƒ'' in the <math>L^1</math> norm. * There is no Gibbs phenomenon. Results about summability can also imply results about regular convergence. For example, we learn that if ''ƒ'' is continuous at ''t'', then the Fourier series of ''ƒ'' cannot converge to a value different from ''ƒ''(''t''). It may either converge to ''ƒ''(''t'') or diverge. This is because, if <math>S_N(f;t)</math> converges to some value ''x'', it is also summable to it, so from the first summability property above, ''x'' = ''ƒ''(''t''). ==Order of growth== The order of growth of Dirichlet's kernel is logarithmic, i.e. :<math>\int |D_N(t)|\,\mathrm{d}t = \frac{4}{\pi^2}\log N+O(1).</math> See [[Big O notation]] for the notation ''O''(1). The actual value <math>4/\pi^2</math> is both difficult to calculate (see Zygmund 8.3) and of almost no use. The fact that for ''some'' constant ''c'' we have :<math>\int |D_N(t)|\,\mathrm{d}t > c\log N+O(1)</math> is quite clear when one examines the graph of Dirichlet's kernel. The integral over the ''n''-th peak is bigger than ''c''/''n'' and therefore the estimate for the [[Harmonic series (mathematics)|harmonic sum]] gives the logarithmic estimate. This estimate entails quantitative versions of some of the previous results. For any continuous function ''f'' and any ''t'' one has :<math>\lim_{N\to\infty} \frac{S_N(f;t)}{\log N}=0.</math> However, for any order of growth ω(''n'') smaller than log, this no longer holds and it is possible to find a continuous function ''f'' such that for some ''t'', :<math>\varlimsup_{N\to\infty} \frac{S_N(f;t)}{\omega(N)}=\infty.</math> The equivalent problem for divergence everywhere is open. Sergei Konyagin managed to construct an integrable function such that for ''every t'' one has :<math>\varlimsup_{N\to\infty} \frac{S_N(f;t)}{\sqrt{\log N}}=\infty.</math> It is not known whether this example is best possible. The only bound from the other direction known is log ''n''.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)