Symmetric derived number

at a point $ x $

A generalization of the ordinary notion of a derived number (cf. Dini derivative) to the case of a set function $ \Phi $ on an $ n $- dimensional Euclidean space. The symmetric derived numbers of $ \Phi $ at $ x $ are defined as the limits

$$ \lim\limits _ {k \rightarrow \infty } \ \frac{\Phi ( S ( x, r _ {k} )) }{| S ( x, r _ {k} ) | } , $$

where $ S ( x, r _ {k} ) $ is some sequence of closed balls with centres at $ x $ and radii $ r _ {k} $ such that $ r _ {k} \rightarrow 0 $ as $ k \rightarrow \infty $.

The $ n $- th symmetric derived numbers at $ x $ of a function $ f $ of a real variable are defined as the limits

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

$$ = \ \lim\limits _ {k \rightarrow \infty } \frac{\sum _ {m = 0 } ^ { n } \left ( \begin{array}{c} n \\ m \end{array} \right ) (- 1) ^ {m} f \left ( x + { \frac{n - 2m }{2} } h _ {k} \right ) }{h _ {k} ^ {n} } , $$

where $ h _ {k} \rightarrow 0 $ as $ k \rightarrow \infty $ and $ \Delta _ {s} ^ {n} f ( x, h _ {k} ) $ is the symmetric difference of order $ n $ of $ f $ at $ x $.

References

[1]	S. Saks, "Theory of the integral" , Hafner (1937) (Translated from French)

References

[a1]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24