Variations of sets and functions.
(Variatsii mnozhestv i funktsij.)

*(Russian)*Zbl 0967.26500
Moscow: Nauka, 352 p. (1975).

From the preface: Variations of a set are metric characteristics arising in a natural way in various problems of analysis. Estimates of the complexity of the computation algorithms of functions lead to a comparison between the set of functions approximated and the set of functions obtained by using the algorithm. If the tolerance error is small, then the second of these two sets cannot be much simpler than the first one. The variations of a set enable us to make precise the above type of statement and thus to compare the efficiency of algorithms. In this way, for example, it was found that application of rational functions instead of polynomials with the same number of parameters does not give any increase in precision. In the same way, it was proved that there exist smooth functions of three variables which are not representable as the composition of smooth functions of two variables.

It is also of interest to study the class of sets whose variations are finite. The \(p\)-dimensional sets of this class have a tangent \(p\)-plane at almost all their points with respect to Hausdorff \(p\)-measure. This makes it possible to prove, for example, that all the known two-dimensional measures on surfaces with finite variations in three-dimensional space are commensurable. What is more, one can prove that all two-dimensional measures are commensurable if they satisfy a rather natural system of axioms.

Variations of a set lead to a generalization, to \(n\) variables, of the classic definition of the variation of a function of one variable. Functions of \(n\) variables with finite variations have many of the properties of functions of one variable of bounded variation. For example, they can be decomposed into the difference of ‘monotone’ functions, they have a total differential almost everywhere, and their Fourier series converge to them almost everywhere.

This book contains a systematic presentation of the theory of variations.

It is also of interest to study the class of sets whose variations are finite. The \(p\)-dimensional sets of this class have a tangent \(p\)-plane at almost all their points with respect to Hausdorff \(p\)-measure. This makes it possible to prove, for example, that all the known two-dimensional measures on surfaces with finite variations in three-dimensional space are commensurable. What is more, one can prove that all two-dimensional measures are commensurable if they satisfy a rather natural system of axioms.

Variations of a set lead to a generalization, to \(n\) variables, of the classic definition of the variation of a function of one variable. Functions of \(n\) variables with finite variations have many of the properties of functions of one variable of bounded variation. For example, they can be decomposed into the difference of ‘monotone’ functions, they have a total differential almost everywhere, and their Fourier series converge to them almost everywhere.

This book contains a systematic presentation of the theory of variations.