Dini test

Definition

Let f be a function on [0,2], 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 and ε is negative then we define .

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 that ::\int_0^\pi \frac{1}{\delta}\omega_f(\delta;t),\mathrm{d}\delta :Then the Fourier series of f converges at t to f(t).

For example, the theorem holds with but does not hold with log−1().

:Theorem (the Dini–Lipschitz test): Assume a function f satisfies ::\omega_f(\delta)=o\left(\log\frac{1}{\delta}\right)^{-1}. :Then the Fourier series of f converges uniformly to f.

In particular, any function that obeys a Hölder condition 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.

:\omega_f(\delta)=O\left(\log\frac{1}{\delta}\right)^{-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

:\int_0^\pi \frac{1}{\delta}\Omega(\delta),\mathrm{d}\delta = \infty

there exists a function f such that

:\omega_f(\delta;0)

and the Fourier series of f diverges at 0.

References

References

1. Gustafson, Karl E.. (1999). "Introduction to Partial Differential Equations and Hilbert Space Methods". Courier Dover Publications.