Riemann derivative

Schwarzian derivative, second symmetric derivative, of a function $ f $ at a point $ x _ {0} $

The limit

$$ D ^ {2} f( x _ {0} ) = \ \lim\limits _ {h \rightarrow 0 } \frac{f( x _ {0} + h) - 2f( x _ {0} ) + f( x _ {0} - h) }{h ^ {2} } . $$

It was introduced by B. Riemann in 1854, who proved that if at a point $ x _ {0} $ the second derivative $ f ^ { \prime\prime } ( x _ {0} ) $ exists, then so does the Riemann derivative and $ D ^ {2} f( x _ {0} ) = f ^ { \prime\prime } ( x _ {0} ) $. The upper and lower limits of

$$ \frac{f( x _ {0} + h) - 2f( x _ {0} ) + f( x _ {0} + h) }{h ^ {2} } $$

as $ h \rightarrow 0 $ are called the upper ( $ {\overline{D}\; } {} ^ {2} f( x _ {0} ) $) and lower ( $ \underline{D} ^ {2} f( x _ {0} ) $) Riemann derivative, respectively.

Riemann derivatives find wide application in the theory of the representation of functions by trigonometric series, and in particular in connection with the Riemann summation method.

References

[a1]	T.M. Apostol, "Mathematical analysis" , Blaisdell (1957)
[a2]	B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , Gesammelte Math. Abhandlungen , Dover, reprint (1957) pp. 227–264
[a3]	J. Wolff, "Fourier'sche Reihen" , Noordhoff (1931)