Skip to main content

On functions of bounded variation


The recently introduced concept of ${\mathcal D}$ -variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from [20] whether every function of bounded Hardy–Krause variation is Borel measurable and has bounded ${\mathcal D}$ -variation. Moreover, we show that the space of functions of bounded ${\mathcal D}$ -variation can be turned into a commutative Banach algebra.

Hide All

Corresponding author: Florian Pausinger, TU Munich, Zentrum Mathematik, M10 Geometrie und Visualisierung, Boltzmannstr. 3, 85748 Garching.

Hide All
[1] Adams, C. R. and Clarkson, J. A. Properties of functions f(x, y) of bounded variation. Trans. Amer. Math. Soc. 36 (1934), 711730.
[2] Aistleitner, C. and Dick, J. Functions of bounded variation, signed measures and a general Koksma–Hlawka inequality. Acta Arith 167 (2015), 143171.
[3] Antosik, P. Study of the continuity of a function of many variables (Russian). Prace Mat. 10 (1966), 101104.
[4] Blümlinger, M. and Tichy, R. F. Topological Algebras of Functions of Bounded Variation I. Manuscripta Math. 65 (1989), 245255.
[5] Blümlinger, M. Topological Algebras of Functions of Bounded Variation II. Manuscripta Math. 65 (1989), 377384.
[6] Brandolini, L., Colzani, L., Gigante, G. and Travaglini, G. On the Koksma–Hlawka inequality. J. Complexity 29 (2013), 158172.
[7] Brandolini, L., Colzani, L., Gigante, G. and Travaglini, G. A Koksma–Hlawka inequality for simplices, Trends in Harmonic Analysis. Springer INdAM Ser. 3 (Springer, Milan, 2013), 3346.
[8] Clarkson, J. A. and Adams, C. R. On definitions of bounded variation for functions of two variables. Trans. Amer. Math. Soc. 35 (1933), 824854.
[9] Drmota, M. and Tichy, R. F. Sequences, discrepancies and applications. Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1997), 1651.
[10] Götz, M. Discrepancy and the error in integration. Monatsh. Math. 136 (2002), no. 2, 99121.
[11] Hardy, G. H. On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Quart. J. Math. (1) 37 (1906), 5379.
[12] Harman, G. Variations on the Koksma–Hlawka inequality. Unif. Distrib. Theory 5 (2010), 6578.
[13] Hlawka, E. Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Math. Pura Appl. 54 (1961), 325333.
[14] Idczak, D. Functions of several variables of finite variation and their differentiability. Ann. Polon. Math. 60 (1994), no. 1, 4756.
[15] Koksma, J. F. A general theorem from the theory of uniform distribution modulo 1. Mathematica B (Zutphen) 11 (1942), 711.
[16] Krause, J. M. Fouriersche Reihen mit zwei veränderlichen Grössen. Ber. Sächs. Akad. Wiss. Leipzig 55 (1903) 164197.
[17] Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences (Wiley-Interscience John Wiley & Sons, New York–London–Sydney, 1974).
[18] Leonov, A. S. Remarks on the total variation of functions of several variables and on a multidimensional analogue of Helly's choice principle. (in Russian) Mat. Zametki 63 (1998), 6980.
[19] Owen, A. B. Multidimensional variation for quasi-Monte Carlo. Contemporary multivariate analysis and design of experiments. Ser. Biostat. 2 (World Scientific Publishing Co. Pte. Ltd., 2005), 4974.
[20] Pausinger, F. and Svane, A. M. A Koksma–Hlawka inequality for general discrepancy systems. J. Complexity 31 (2015), 773797.
[21] Yeh, J. Real analysis. Theory of measure and integration. Second edition (World Scientific Publishing Co. Pte. Ltd., 2006).
[22] Young, W. H. and Young, G. On the discontinuties of monotone functions of several variables. Proc. London Math. Soc. (2) 22 (1923), 124142.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *