Difference between revisions of "Lipschitz condition"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.P. Natanson, "Constructive function theory" , '''1–3''' , F. Ungar (1964–1965) (Translated from Russian)</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988) {{MR|0933759}} {{ZBL|0628.42001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.P. Natanson, "Constructive function theory" , '''1–3''' , F. Ungar (1964–1965) (Translated from Russian) {{MR|1868029}} {{MR|0196342}} {{MR|0196341}} {{MR|0196340}} {{ZBL|1034.01022}} {{ZBL|0178.39701}} {{ZBL|0136.36302}} {{ZBL|0133.31101}} </TD></TR></table> |
Revision as of 12:12, 27 September 2012
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) MR0933759 Zbl 0628.42001 |
[3] | I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) MR1868029 MR0196342 MR0196341 MR0196340 Zbl 1034.01022 Zbl 0178.39701 Zbl 0136.36302 Zbl 0133.31101 |
Lipschitz condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_condition&oldid=14093