×

Binomial measures and their approximations. (English) Zbl 1245.28001

Let \(0<\alpha<1\). It is first proved that there exists a unique measure \(\mu_\alpha\) on the Borel algebra \(\mathcal{B}\) of \([0,1]\) such that \(\mu_\alpha(E)=(1-\alpha)\mu_\alpha S_1^{-1}(E)+\alpha\mu_\alpha S_2^{-1}(E)\) for every \(E\in\mathcal{B}\) where \(S_1=x/2\) and \(S_2=x/2+1/2\). The authors study several properties of these measures \(\mu_\alpha\), called here binomial measures. In particular it is proved that the transformation of \(([0,1,\mathcal{B},\mu_\alpha)\) given by \(T(x):=2x-[2x]\) is measure preserving and strongly mixing, i.e., \(\lim_{n\rightarrow\infty}\nu(T^{-n}(B_1)\cap B_2)=\nu(B_1)\nu(B_2)\) whenever \(B_1,B_2\in\mathcal{B}\). Moreover, the authors deal with error estimations when approximating \(\int_0^1f(x)d\mu_\alpha\) by quadrature rules used in numerical analysis.
Reviewer: Hans Weber (Udine)

MSC:

28A25 Integration with respect to measures and other set functions
28A80 Fractals
65D32 Numerical quadrature and cubature formulas
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] M.F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals , Proc. Roy. Soc. London Ser. A 399 (1857) (1985), 243-275. · Zbl 0588.28002 · doi:10.1098/rspa.1985.0057
[2] F. Calabrò and A. Corbo Esposito, An efficient and reliable quadrature algorithm for integration with respect to binomial measures , BIT 48 (3) (2008), 473-491. · Zbl 1155.65022 · doi:10.1007/s10543-008-0168-x
[3] F. Calabrò and A. Corbo Esposito, An evaluation of Clenshaw-Curtis quadrature rule for integration w.r.t. singular measures , J. Comput. Appl. Math. 229 (1) (2009), 120- 128. · Zbl 1166.65010 · doi:10.1016/j.cam.2008.10.022
[4] L. Carbone, G. Cardone, and A. Corbo Esposito, Binary digits expansion of numbers: Hausdorff dimensions of intersections of level sets of avarages’ upper and lower limits , Scientiae Mathematicae Japonicae 60 (2) (2004), 347-356. · Zbl 1073.28005
[5] Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications , second ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998. · Zbl 0896.60013
[6] C.J.G. Evertsz and B.B. Mandelbrot, Multifractal measures , Chaos and Fractals, ch. Appendix B, 921-953. Springer-Verlag, 1992.
[7] W. Gautschi, Orthogonal polynomials and quadrature , ETNA 9 (1999), 65-76. · Zbl 0963.33004
[8] J.E. Hutchinson, Fractals and self-similarity , Indiana Univ. Math. J. 30 (5) (1981), 713-747. · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[9] Arnold R. Krommer and Christoph W. Ueberhuber, Computational integration , Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics, 1998. · Zbl 0903.65019
[10] D.P. Laurie and J.M. de Villiers, Orthogonal polynomials for refinable linear functionals , Math. Comp. 75 (256) (2006), 1891-1903 (electronic). · Zbl 1107.65130 · doi:10.1090/S0025-5718-06-01855-2
[11] G. Mantica, A stable Stieltjes technique for computing orthogonal polynomials and Jacobi matrices associated with a class of singular measures , Constr. Approx. 12 (4) (1996), 509-530. · Zbl 0878.42014 · doi:10.1007/BF02437506
[12] G. Mantica, Fractal measures and polynomial sampling: IFS-Gaussian integration , Numer. Algor. 45 (1-4) (2007), 269-281. · Zbl 1131.65027 · doi:10.1007/s11075-007-9111-5
[13] Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples , Chaos 7 (1) (1997), 89-106. · Zbl 0933.37020 · doi:10.1063/1.166242
[14] Y. Pesin and H. Weiss, Global analysis of dynamical systems , ch. The Maltifractal Analysis of Birkhoff Averages and Large Deviations, IoP Publishing, Bristol, 2001. · Zbl 0996.37021
[15] A. Quarteroni, R. Sacco, and F. Saleri, Numerical mathematics , second ed., Texts in Applied Mathematics, vol. 37, Springer-Verlag, Berlin, 2007. · Zbl 1136.65001
[16] R.H. Riedi, “Multifractal processes” in Theory and applications of long-range dependence , Birkhäuser, ISBN: 0817641688. Switzerland, 2002, 625-715.
[17] L.N. Trefethen, Is Gauss quadrature better than Clenshaw Curtis? , SIAM Review 50 (1) (2008), 67-87. · Zbl 1141.65018 · doi:10.1137/060659831
[18] P. Walters, An introduction to ergodic theory , Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York, 1982. · Zbl 0475.28009
[19] Andrew Q. Yingst, A characterization of homeomorphic Bernoulli trial measures , Trans. Amer. Math. Soc. 360 (2) (2008), 1103-1131 (electronic). · Zbl 1147.28008 · doi:10.1090/S0002-9947-07-04431-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.