Mutually singular functions and computation of the lengths of curves

A. A. Dovgoshey ; O. Martio

Izvestiya. Mathematics , Tome 70 (2006) no. 4, pp. 693-716.

Résumé

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.

Bibliographie

[1] Pelling M. J., “Formulae for the arc-length of a curve in $\mathbb R^N$”, Amer. Math. Monthly, 84:6 (1977), 465–467 | DOI | MR | Zbl

[2] Thomson B. S., Real functions, Lecture Notes in Math., 1170, Springer-Verlag, Berlin etc., 1985 | MR | Zbl

[3] Thomson B. S., “Singular functions”, Rev. Roumaine Math. Pures Appl., 30:9 (1985), 793–800 | MR | Zbl

[4] Garg K. M., “Derivatives of variation functions and of mutually singular and relatively absolutely continuous functions”, Real Anal. Exchange, 10 (1984–1985), 54–57 | MR | Zbl

[5] Garg K. M., “Characterizations of absolutely continuous and singular functions”, Proc. of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969), Akademiai Kaido, Budapest, 1972, 183–188 | MR | Zbl

[6] Garg K. M., “On singular functions”, Rev. Roumaine Math. Pures Appl., 14:10 (1969), 1441–1452 | MR | Zbl

[7] Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987 | MR | Zbl

[8] Mattila P., Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[9] Ambrosio L., Tilli P., Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, 25, Cambridge University Press, Cambridge, 2004 | MR | Zbl

[10] Rado T., Reichelderfer P. V., Continuous transformation in analysis with an introduction to algebraic topology, Grundlehren der mathematischen Wissenschaften, LXXV, Springer-Verlag, Berlin etc., 1955 | MR | Zbl

[11] Evans L. C., Gariepy R. F., Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Rato–London, 1992 | MR | Zbl

[12] Bruckner A. M., Differentiation of real functions, Lecture Notes in Math., 659, Springer-Verlag, Berlin etc., 1978 | MR | Zbl

[13] Dovgoshey O., Martio O., “Singular functions and arc length”, Reports in Math., 394, University of Helsinki, 2004, 15

[14] Fedorchuk V. V., Filippov V. V., Obschaya topologiya. Osnovnye konstruktsii, Izd-vo MGU, M., 1988 | Zbl

[15] Buzeman G., Vypuklye poverkhnosti, Nauka, M., 1964 | MR | Zbl

[16] Gryunbaum B., “Problema Borsuka i rodstvennye ei voprosy”, Etyudy po kombinatornoi geometrii i teorii vypuklykh tel, Nauka, M., 1971 | MR | Zbl

[17] Caratheodory C., “Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen”, Math. Ann., 64:1 (1907), 95–115 | DOI | MR | Zbl

[18] Caratheodory C., “Über den Variabilitätsbereich der Fourierscher Konstanten for positiven harmonishen Funktionen”, Rend. Circ. Mat. Palermo, 32 (1911), 193–217 | DOI | Zbl

[19] Doob J. L., Measure theory, Graduate Texts in Mathematics, 143, Springer-Verlag, New York, 1994 | MR | Zbl

[20] Dovgoshey O., Martio O., “The arc length evaluations and Lebesgue decomposition”, Reports of the Dep. of Math., 379, University of Helsinki, 2004, 11