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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.

Definition

Let f be a function on , 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 ) | {\displaystyle \left.\right.\omega _{f}(\delta ;t)=\max _{|\varepsilon |\leq \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

ω f ( δ ) = max t ω f ( δ ; t ) {\displaystyle \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
∫ 0 π 1 δ ω f ( δ ; t ) d δ < ∞ . {\displaystyle \int _{0}^{\pi }{\frac {1}{\delta }}\omega _{f}(\delta ;t)\,\mathrm {d} \delta <\infty .}
Then the Fourier series of f converges at t to f(t).

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

Theorem (the Dini–Lipschitz test): Assume a function f satisfies
ω f ( δ ) = o ( log ⁡ 1 δ ) − 1 . {\displaystyle \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.

ω f ( δ ) = O ( log ⁡ 1 δ ) − 1 . {\displaystyle \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

∫ 0 π 1 δ Ω ( δ ) d δ = ∞ {\displaystyle \int _{0}^{\pi }{\frac {1}{\delta }}\Omega (\delta )\,\mathrm {d} \delta =\infty }

there exists a function f such that

ω f ( δ ; 0 ) < Ω ( δ ) {\displaystyle \omega _{f}(\delta ;0)<\Omega (\delta )}

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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