Difference between revisions of "De la Vallée-Poussin derivative"
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Revision as of 18:51, 24 March 2012
generalized symmetric derivative
A derivative defined by Ch.J. de la Vallée-Poussin [1]. Let be an even number and let there exist a
such that for all
with
,
![]() | (*) |
![]() |
where are constants,
as
and
. The number
is called the de la Vallée-Poussin derivative of order
, or the symmetric derivative of order
, of the function
at the point
.
The de la Vallée-Poussin derivatives of odd orders are defined in a similar manner, equation (*) being replaced by
![]() |
![]() |
The de la Vallée-Poussin derivative is identical with Riemann's second derivative, often called the Schwarzian derivative. If
exists,
,
, also exist, but
need not exist. If there exists a finite ordinary two-sided derivative
, then
. For the function
, for example,
,
and the
,
do not exist. If there exists a de la Vallée-Poussin derivative
, the series
obtained from the Fourier series of
by term-by-term differentiation repeated
times is summable at
to
by the method
for
, [2] (cf. Cesàro summation methods).
References
[1] | Ch.J. de la Vallée-Poussin, "Sur l'approximation des fonctions d'une variable reélle et de leurs dériveés par des polynômes et des suites limiteés de Fourier" Bull. Acad. Belg. , 3 (1908) pp. 193–254 |
[2] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) pp. Chapt.11 |
De la Vallée-Poussin derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_derivative&oldid=22325