Dini test

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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 converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz.[1]

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 ωf = log−2(1/δ) but does not hold with log−1(1/δ).

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 Hölder class[clarification needed] satisfies the Dini–Lipschitz test.

Precision

Both tests are the 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

References

  1. ^ Gustafson, Karl E. (1999), Introduction to Partial Differential Equations and Hilbert Space Methods, Courier Dover Publications, p. 121, ISBN 978-0-486-61271-3 
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