Lipschitz condition
A restriction on the behaviour of increase of a function. If for any points and
belonging to an interval
the increase of a function
satisfies the inequality
![]() | (*) |
where and
is a constant, then one says that
satisfies a Lipschitz condition of order
on
and writes
,
or
. Every function that satisfies a Lipschitz condition with some
on
is uniformly continuous on
, and functions that satisfy a Lipschitz condition of order
are absolutely continuous (cf. Absolute continuity; Uniform continuity). A function that has a bounded derivative on
satisfies a Lipschitz condition on
with any
.
The Lipschitz condition (*) is equivalent to the condition
![]() |
where is the modulus of continuity (cf. Continuity, modulus of) of
on
. Lipschitz conditions were first considered by R. Lipschitz [1] as a sufficient condition for the convergence of the Fourier series of
. In the case
the condition (*) is also called a Hölder condition of order
.
References
[1] | R. Lipschitz, "De explicatione per series trigonometricas insttuenda functionum unius variablis arbitrariarum, et praecipue earum, quae per variablis spatium finitum valorum maximorum et minimorum numerum habent infintum disquisitio" J. Reine Angew. Math. , 63 (1864) pp. 296–308 |
[2] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |
[3] | I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) |
Lipschitz condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_condition&oldid=14093