Divided differences, square functions, and a law of the iterated logarithm. (English) Zbl 1408.26004

Various integral means of symmetric divided differences of the 1st and 2nd rank, i.e., \[ \Delta(f)(x,t)=\frac{f(x+t)-f(x-t)}{2t},\qquad\Delta_2(f)(x,t)=\frac{f(x+t)+f(x-t)-2f(x)}{2t},\tag{\(\ast\)} \] are used as a tool to characterise the set of differentiability points of a function \(f\) on an interval, and, as well, \(f\) as a member of the Sobolev space \(W^{1,p}\). After establishing the one-dimensional results, Theorems 2 and 3, with the means of (\(\ast\)) generalised in terms of directional derivatives, corresponding multi-variable counterparts are proved.


26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
60G46 Martingales and classical analysis
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