Symmetric derivative

A generalization of the concept of derivative to the case of set functions $ \Phi $ on an $ n $- dimensional Euclidean space. The symmetric derivative at a point $ x $ is the limit

$$ \lim\limits _ {r \downarrow 0 } \ \frac{\Phi ( S ( x; r)) }{| S ( x; r) | } \equiv \ D _ { \mathop{\rm sym} } \Phi ( x), $$

where $ S ( x; r) $ is the closed ball with centre $ x $ and radius $ r $, if this limit exists. The symmetric derivative of order $ n $ at a point $ x $ of a function $ f $ of a real variable is defined as the limit

$$ \lim\limits _ {h \rightarrow 0 } \ \frac{\Delta _ {s} ^ {n} f ( x, h) }{h ^ {n} } = $$

$$ = \ \lim\limits _ {h \rightarrow 0 } \frac{\sum _ {k = 0 } ^ { n } \left ( \begin{array}{c} n \\ k \end{array} \right ) (- 1) ^ {k} f \left ( x + { \frac{n - 2k }{2} } h \right ) }{h ^ {n} } = D _ { \mathop{\rm sym} } ^ {n} f ( x). $$

A function $ f $ of a real variable has a symmetric derivative of order $ 2r $ at a point $ x $,

$$ D _ { \mathop{\rm sym} } ^ {2r} f ( x) = \beta _ {2r} , $$

if

$$ { \frac{1}{2} } ( f ( x + h) + f ( x - h)) - \sum _ {k = 0 } ^ { r } \beta _ {2k} \frac{h ^ {2k} }{( 2k)! } = \ o ( h ^ {2r} ); $$

and one of order $ 2r + 1 $,

$$ D _ { \mathop{\rm sym} } ^ {2r + 1 } f ( x) = \ \beta _ {2r + 1 } , $$

if

$$ { \frac{1}{2} } ( f ( x + h) - f ( x - h)) - \sum _ {k = 0 } ^ { r } \beta _ {2k + 1 } \frac{h ^ {2k + 1 } }{( 2k + 1)! } = \ o ( h ^ {2r + 1 } ). $$

If $ f $ has an $ n $- th order derivative $ f ^ { ( n) } $ at a point $ x $, then there is (in both cases) a symmetric derivative at $ x $, and it is equal to $ f ^ { ( n) } ( x) $.

References

[1]	S. Saks, "Theory of the integral" , Hafner (1937) (Translated from French)
[2]	R.D. James, "Generalized th primitives" Trans. Amer. Math. Soc. , 76 : 1 (1954) pp. 149–176

Comments

In [1] instead of derivative, "derivate" is used: symmetric derivate.