De la Vallée-Poussin derivative

generalized symmetric derivative

A derivative defined by Ch.J. de la Vallée-Poussin [1]. Let $ r $ be an even number and let there exist a $ \delta > 0 $ such that for all $ t $ with $ | t | < \delta $,

$$ \tag{* } { \frac{1}{2} } \{ f ( x _ {0} + t) + f ( x _ {0} - t) \} = $$

$$ = \ \beta _ {0} + \frac{t ^ {2} \beta _ {2} }{2 } + \dots + \frac{t ^ {r} \beta _ {r} }{r! } + \gamma ( t) t ^ {r} , $$

where $ \beta _ {0} \dots \beta _ {r} $ are constants, $ \gamma ( t) \rightarrow 0 $ as $ t \rightarrow 0 $ and $ \gamma ( 0) = 0 $. The number $ \beta _ {r} = f _ {(} r) ( x _ {0} ) $ is called the de la Vallée-Poussin derivative of order $ r $, or the symmetric derivative of order $ r $, of the function $ f $ at the point $ x _ {0} $.

The de la Vallée-Poussin derivatives of odd orders $ r $ are defined in a similar manner, equation (*) being replaced by

$$ { \frac{1}{2} } \{ f ( x _ {0} + t) - f ( x _ {0} - t) \} = $$

$$ = \ t \beta _ {1} + \frac{t ^ {3} \beta _ {3} }{ 3! } + \dots + \frac{t ^ {r} \beta _ {r} }{r! } + \gamma ( t) t ^ {r} . $$

The de la Vallée-Poussin derivative $ f _ {(} 2) ( x _ {0} ) $ is identical with Riemann's second derivative, often called the Schwarzian derivative. If $ f _ {(} r) ( x _ {0} ) $ exists, $ f _ {(} r- 2) ( x _ {0} ) $, $ r \geq 2 $, also exist, but $ f _ {(} r- 1) ( x _ {0} ) $ need not exist. If there exists a finite ordinary two-sided derivative $ f ^ { ( r) } ( x _ {0} ) $, then $ f _ {(} r) ( x _ {0} ) = f ^ { ( r) } ( x _ {0} ) $. For the function $ f( x) = { \mathop{\rm sgn} } x $, for example, $ f _ {(} 2k) ( 0) = 0 $, $ k = 1, 2 \dots $ and the $ f _ {(} 2k+ 1) ( 0) $, $ k = 0, 1 \dots $ do not exist. If there exists a de la Vallée-Poussin derivative $ f _ {(} r) ( x _ {0} ) $, the series $ S ^ { ( r) } ( f ) $ obtained from the Fourier series of $ f $ by term-by-term differentiation repeated $ r $ times is summable at $ x _ {0} $ to $ f _ {(} r) ( x _ {0} ) $ by the method $ ( C, \alpha ) $ for $ \alpha > r $, [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