A formula for the derivative in terms of finite differences (Q689761)

Let \(E\) be a normed vector space of \(\mathbb{R}\) and \(F\) a real Banach space. For \(x_ 0\) in \(E\) and \(r>0\) let \(B'(x_ 0,r)=\{x\in E:\| x-x_ 0\|\leq r\}\) denote the closed ball of center \(x_ 0\) and radius \(r\). Given a function \(f\) defined on a neighbourhood of \(x_ 0\) with values in \(R\), we let: \[ \Delta^ 1 f(x_ 0;h)=f(x_ 0+h)-f(x_ 0), \qquad \Delta^ 2 f(x_ 0;h_ 1,h_ 2)+ \Delta^ 1 f(x_ 0+ h_ 2;h_ 1)- \Delta^ 1 f(x_ 0;h_ 1). \] Proposition. Let the map \(f\) have the following properties: (i) \(f\) is bounded near \(x_ 0\) (say in \(B'(x_ 0,2r))\); (ii) there are two constants \(\alpha>0\) and \(\beta>0\) subject to the condition \(\alpha+\beta>1\) such that the ratio \({{\Delta^ 2 f(x_ 0;h_ 1,h_ 2)} \over {\| h_ 1\|^ \alpha \| h_ 2\|^ \beta}}\) remains bounded whenever the nonzero vectors \(h_ 1\) and \(h_ 2\) are near 0 (say in \(B'(0,r))\). Then the function \(f\) is Fréchet differentiable at \(x_ 0\) and its Fréchet derivative is given by the formula \[ Df(x_ 0)(h)= \Delta^ 1 f(x_ 0;h)- \sum^ \infty_{i=1} 2^{i-1} \Delta^ 2 f\left( x_ 0; {h \over {2^ i}}, {h \over {2^ i}}\right), \qquad \| h\|\leq r. \] The proposition extends, in the context of Banach spaces, a lemma of \textit{Jean Dieudonne} [Ann. Sci. Ecole Normale Sup., III. Ser. 61, 231-248 (1944; Zbl 0060.138)], under weaker conditions. The author gives a new proof of this lemma and establishes a formula for the derivative in terms of finite differences. Remarks. 1) The map \(f\) is lipschitzian of order \(\gamma=\min\{1,\max(\alpha,\beta)\}\); 2) the conclusion of the theorem remains valid if we replace condition (ii) by the following werker condition: (ii') there exists \(\beta>1\) and \(M\geq 0\) such that \(\|\Delta^ 2 f(x_ 0;h,h)\|\leq M\| h\|^ \beta\) for all \(h\) in \(B'(0,r)\).