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Concept# Dini test

Summary

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.
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
:\left.\right.\omega_f(\delta;t)=\max_{|\varepsilon| \le \delta} |f(t)-f(t+\varepsilon)|
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
:\omega_f(\delta) = \max_t \omega_f(\delta;t)
With these definitions we may state the main results:
:Theorem (Dini's test): Assume a function f satisfies at a point t

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