Dini test

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

ω f ( δ ; t ) = max | ε | ≤ δ | f ( t ) − f ( t + ε ) |

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

ω f ( δ ) = max t ω f ( δ ; t )

With these definitions we may state the main results:

Theorem (Dini's test): Assume a function f satisfies at a point t that ∫ 0 π 1 δ ω f ( δ ; t ) d δ < ∞ . Then the Fourier series of f converges at t to f(t).

For example, the theorem holds with ω_f = log⁻²(1/δ) but does not hold with log⁻¹(1/δ).

Theorem (the Dini–Lipschitz test): Assume a function f satisfies ω f ( δ ) = o ( log ⁡ 1 δ ) − 1 . Then the Fourier series of f converges uniformly to f.

In particular, any function of a Hölder class 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.

ω f ( δ ) = O ( log ⁡ 1 δ ) − 1 .

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

∫ 0 π 1 δ Ω ( δ ) d δ = ∞

there exists a function f such that

ω f ( δ ; 0 ) < Ω ( δ )

and the Fourier series of f diverges at 0.