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Dini TestIn mathematics, the Dini and Dini-Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz. Definition 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 -
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). The global modulus of continuity (or simply the modulus of continuity) is defined by -
With these definitions we may state the main results Theorem (Dini's test): Assume a function f satisfies at a point t that -
Then the Fourier series of f converges at t to f(t). For example, the theorem holds with but does not hold with . Theorem (the Dini-Lipschitz test): Assume a function f satisfies -
Then the Fourier series of f converges uniformly to f. In particular, any function of a Hlder class satisfies the Dini-Lipschitz test. Precision 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. -
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 -
there exists a function f such that -
and the Fourier series of f diverges at 0. See also
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