De la Vallée-Poussin derivative

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