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