The book under review is a remarkable treatise on the theory of functions of bounded variation. This notion was introduced by Camille Jordan in 1881 under the name functions of limited oscillations. His goal was to linearize Dirichlet's condition for the convergence of Fourier series. Let us recall Jordan's definition: "Let x_1,..., x_n be a series of values of x between 0 and ∈, and y_1,...,y_n the corresponding values of f(x). The points x_1, y_1; ...; x_n, y_n will form a polygon. Consider the differences y_2-y_1, y_3-y_2,...,y_n-y_(n-1). We will call the sum of the positive terms of this sequence the positive oscillation of the polygon; negative oscillation is the sum of the negative terms; total oscillation is the sum of those two partial oscillations in absolute value. Let us vary the polygon; two cases may occur: 1° The polygon may be chosen so that its oscillations exceed every limit. 2° For every chosen polygon, its positive and negative oscillations will be less than some fixed limits P_∈ and N_∈. We will say in that case that F(x) is a function of limited oscillation in the interval from 0 to ∈; P_∈ will be its positive oscillation; N_∈ its negative oscillation; P_∈ + N_∈ its total oscillation".
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