Difference between revisions of "Dini-Lipschitz criterion"
From Encyclopedia of Mathematics
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| − | + | ''for the convergence of Fourier series'' | |
| − | + | {{MSC|42A20}} | |
| − | + | {{TEX|done}} | |
| − | + | [[Category:Harmonic analysis on Euclidean spaces]] | |
| − | + | A criterion proved independently by Dini and Lipschitz for the uniform convergence of [[Trigonometric series|Fourier series]], see {{Cite|Di}} and {{Cite|Li}}. | |
| + | |||
| + | Consider a continuous function $f:{\mathbb R} \to {\mathbb R}$ which is $2\pi$-periodic and denote by $\omega (\delta, I)$ its modulus of continuity, namely | ||
| + | \[ | ||
| + | \omega (\delta, I) := \sup\; \{|f(x)-f(y)| : x,y\in I \;\mbox{and}\; |x-y|\leq \delta\}\, . | ||
| + | \] | ||
| + | The Dini-Lipschitz criterion is then the following theorem (cp. with Theorems 10.3 and 10.5 of {{Cite|Zy}}): | ||
| + | |||
| + | '''Theorem 1''' | ||
| + | If on some ''open'' interval $I$ we have | ||
| + | \[ | ||
| + | \lim_{\delta\to 0}\; \omega (\delta, I) |\log \delta| = 0\, | ||
| + | \] | ||
| + | then the Fourier series of $f$ converges uniformly to $f$ on any ''closed'' interval $J\subset I$. | ||
| + | |||
| + | Note that, as an obvious corollary, if the interval $I$ has length larger than $2\pi$, then the Fourier series converges uniformly to $f$ on the entire real axis. | ||
| + | |||
| + | The Dini-Lipschitz criterion is sharp in the following sense. If $f: {\mathbb R}^+\to {\mathbb R}^+$ is any function such that | ||
| + | \[ | ||
| + | \limsup_{\delta\to 0}\; f (\delta) |\log \delta| > 0\, | ||
| + | \] | ||
| + | then there is a continuous function $f$ such that $\omega (\delta, {\mathbb R}) \leq f(\delta)$ for any $\delta$ and the corresponding Fourier series ''diverges'' at some point. | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Ba}}|| N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964. | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Di}}|| U. Dini, "Sopra la serie di Fourier" , Pisa (1872). | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Le}}|| H. Lebesgue, "Sur la répresentation trigonométrique approchée des fonctions satisfiasants à une condition de Lipschitz" ''Bull. Soc. Math. France'' , '''38''' (1910) pp. 184-210 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Li}}|| R. Lipschitz, "De explicatione per series trigonometricas instituenda functionum unius variabilis arbitrariarum, etc." ''J. Reine Angew. Math.'' , '''63''' : 2 (1864) pp. 296-308 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ni}}|| S.M. Nikol'skii, "On the Dini–Lipschitz condition for convergence of Fourier series" ''Doklady Akad. Nauk SSSR'' , '''73''' : 3 (1950) pp. 457–460 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Zy}}|| A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988) {{MR|0933759}} {{ZBL|0628.42001}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 12:49, 6 October 2012
for the convergence of Fourier series
2020 Mathematics Subject Classification: Primary: 42A20 [MSN][ZBL]
A criterion proved independently by Dini and Lipschitz for the uniform convergence of Fourier series, see [Di] and [Li].
Consider a continuous function $f:{\mathbb R} \to {\mathbb R}$ which is $2\pi$-periodic and denote by $\omega (\delta, I)$ its modulus of continuity, namely \[ \omega (\delta, I) := \sup\; \{|f(x)-f(y)| : x,y\in I \;\mbox{and}\; |x-y|\leq \delta\}\, . \] The Dini-Lipschitz criterion is then the following theorem (cp. with Theorems 10.3 and 10.5 of [Zy]):
Theorem 1 If on some open interval $I$ we have \[ \lim_{\delta\to 0}\; \omega (\delta, I) |\log \delta| = 0\, \] then the Fourier series of $f$ converges uniformly to $f$ on any closed interval $J\subset I$.
Note that, as an obvious corollary, if the interval $I$ has length larger than $2\pi$, then the Fourier series converges uniformly to $f$ on the entire real axis.
The Dini-Lipschitz criterion is sharp in the following sense. If $f: {\mathbb R}^+\to {\mathbb R}^+$ is any function such that \[ \limsup_{\delta\to 0}\; f (\delta) |\log \delta| > 0\, \] then there is a continuous function $f$ such that $\omega (\delta, {\mathbb R}) \leq f(\delta)$ for any $\delta$ and the corresponding Fourier series diverges at some point.
References
| [Ba] | N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964. |
| [Di] | U. Dini, "Sopra la serie di Fourier" , Pisa (1872). |
| [Le] | H. Lebesgue, "Sur la répresentation trigonométrique approchée des fonctions satisfiasants à une condition de Lipschitz" Bull. Soc. Math. France , 38 (1910) pp. 184-210 |
| [Li] | R. Lipschitz, "De explicatione per series trigonometricas instituenda functionum unius variabilis arbitrariarum, etc." J. Reine Angew. Math. , 63 : 2 (1864) pp. 296-308 |
| [Ni] | S.M. Nikol'skii, "On the Dini–Lipschitz condition for convergence of Fourier series" Doklady Akad. Nauk SSSR , 73 : 3 (1950) pp. 457–460 |
| [Zy] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001 |
How to Cite This Entry:
Dini-Lipschitz criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dini-Lipschitz_criterion&oldid=22357
Dini-Lipschitz criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dini-Lipschitz_criterion&oldid=22357
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article