Dini test

Last updated March 25, 2019

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.^[1]

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the weighted sum of a set of simple oscillating functions, namely sines and cosines.

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

Definition

Let $f$ be a function on [0,2 $π$ ], 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 |\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

In mathematical analysis, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the uniform continuity of functions. So, a function f : I → R admits ω as a modulus of continuity if and only if

\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 <\infty .

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

\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 of a Hölder class ^{[ clarification needed ]} 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.

Big O notation notation to describe the limiting behavior of a function

Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation.

\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)<\Omega (\delta )

and the Fourier series of $f$ diverges at 0.

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References

↑ 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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[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

Dini test

Contents

Definition

Precision

See also

Related Research Articles

References