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| In [[mathematics]], the '''Dini''' and '''Dini-Lipschitz tests''' are highly precise tests that can be used to prove that the [[Fourier series]] of a [[function (mathematics)|function]] converges at a given point. These tests are named after [[Ulisse Dini]] and [[Rudolf Lipschitz]].<ref>{{citation|title=Introduction to Partial Differential Equations and Hilbert Space Methods|author= Karl E. Gustafson|year=1999|publisher=Courier Dover Publications|pages=121 |url=http://books.google.com/?id=uu059Rj4x8oC&pg=PA121&dq=%22Dini+test%22|isbn=978-0-486-61271-3}}</ref>
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| == Definition ==
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| Let ''f'' be a function on [0,2π], let ''t'' be some point and let δ be a positive number. We define the '''local modulus of continuity''' at the point ''t'' by
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| :<math>\left.\right.\omega_f(\delta;t)=\max_{|\varepsilon| \le \delta} |f(t)-f(t+\varepsilon)|</math>
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| Notice that we consider here ''f'' to be a periodic function, e.g. if ''t'' = 0 and ε is negative then we ''define'' ''f''(ε) = ''f''(2π + ε).
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| The '''global modulus of continuity''' (or simply the [[modulus of continuity]]) is defined by
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| :<math>\left.\right.\omega_f(\delta) = \max_t \omega_f(\delta;t)</math>
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| With these definitions we may state the main results
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| ''Theorem (Dini's test): Assume a function f satisfies at a point t that''
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| :<math>\int_0^\pi \frac{1}{\delta}\omega_f(\delta;t)\,d\delta < \infty.</math>
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| ''Then the Fourier series of f converges at t to f(t).''
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| For example, the theorem holds with <math>\omega_f=\log^{-2}(\delta^{-1})</math> but does not hold with <math>\log^{-1}(\delta^{-1})</math>.
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| ''Theorem (the Dini-Lipschitz test): Assume a function f satisfies''
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| :<math>\omega_f(\delta)=o\left(\log\frac{1}{\delta}\right)^{-1}.</math>
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| ''Then the Fourier series of f converges uniformly to f.''
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| In particular, any function of a [[Hölder class]]{{clarify|date=July 2010|reason=Target of redirect does not define the term as used here}} satisfies the Dini-Lipschitz test.
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| ==Precision==
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| Both tests are best of their kind. For the Dini-Lipschitz test, it is possible to construct a function ''f'' with its modulus of continuity satisfying the test with ''O'' instead of ''o'', i.e.
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| :<math>\omega_f(\delta)=O\left(\log\frac{1}{\delta}\right)^{-1}.</math>
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| and the Fourier series of ''f'' diverges. For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that
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| :<math>\int_0^\pi \frac{1}{\delta}\Omega(\delta)\,d\delta = \infty</math>
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| there exists a function ''f'' such that
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| :<math>\left.\right.\omega_f(\delta;0) < \Omega(\delta)</math>
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| and the Fourier series of ''f'' diverges at 0.
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| ==See also==
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| * [[Convergence of Fourier series]]
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| * [[Dini continuity]]
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| ==References==
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| {{reflist}}
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| [[Category:Fourier series]]
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| [[Category:Convergence tests]]
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