Mutually singular functions and computation of the lengths of curves

A. A. Dovgoshey^a, O. Martio<sup>b

a</sup> Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
^b University of Helsinki, Department of Mathematics and Statistics

Abstract: We study rectifiable curves given by mutually singular coordinate functions in finite-dimensional normed spaces. We describe these curves in terms of the behaviour of approximative tangents and find a simple formula for their lengths. We deduce from these results new necessary and sufficient conditions for the mutual singularity of finitely many functions of bounded variation.

UDC: 517.52+517.98

MSC: 26B15, 28A75

Received: 11.05.2005

DOI: 10.4213/im715

 English version: Izvestiya: Mathematics, 2006, 70:4, 693–716

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