Schwarz symmetric derivative

of a function $ f $ at a point $ x _ {0} $

The value

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

It is sometimes called the Riemann derivative or the second symmetric derivative. For the first time introduced by B. Riemann in 1854 (see [2]); it was studied by H.A. Schwarz [1]. More generally, the symmetric derivative of order $ n $ is also called a Schwarz symmetric derivative:

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

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

References

[1]	H.A. Schwarz, "Beweis eines für die Theorie der trigonometrischen Reihen in Betracht kommenden Hülfssatzes" , Gesammelte Math. Abhandlungen , Chelsea, reprint (1972) pp. 341–343
[2]	B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" H. Weber (ed.) , B. Riemann's Gesammelte Mathematische Werke , Dover, reprint (1953) pp. 227–271 ((Original: Göttinger Akad. Abh. (1868)))
[3]	I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian)
[4]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[5]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)

Comments

The name general derivative is also used for this notion. A natural approach is to start with the central difference $ f( x _ {0} + h/2 ) - f( x _ {0} - h/2) $, and to define the first symmetric derivative as

$$ Df( x _ {0} ) = \lim\limits _ {h \rightarrow 0 } \ \frac{f( x _ {0} + h/2)- f( x _ {0} - h/2) }{h} = \ \lim\limits _ {h \rightarrow 0 } \ \frac{\Delta _ {h} f ( x _ {0} ) }{h} , $$

and then $ D ^ {n} = D( D ^ {n-} 1 ) $, $ n \geq 1 $, $ D ^ {0} f = f $.

References

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