Moments of the weighted Cantor measures

• Autorzy: Harding Steven N. , Riasanovsky Alexander W. N.

Identyfikatory

DOI

10.1515/dema-2019-0026

• Języki publikacji: EN

Abstrakty

EN

Based on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying [wzór] where φn : x ↦ (x+n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L2μα[0,1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xmdμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the firstmmoments within uniform error ε in O((loglog(1/ε))·m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2,1/2].

Słowa kluczowe

EN

Cantor; moments; orthogonal polynomials; generating function; iterated function system

PL

Cantor; momenty; wielomiany ortogonalne; funkcja generująca; system funkcji iterowanych

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 256--273
• Opis fizyczny: Bibliogr. 10 poz., wykr.

Twórcy

autor

Harding Steven N.

• Iowa State University, Department of Mathematics, 411 Morrill Road, Ames, IA50011, U.S.A.

autor

Riasanovsky Alexander W. N.

• Iowa State University, Department of Mathematics, 411 Morrill Road, Ames, IA50011, U.S.A.

Bibliografia

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• [3] Jorgensen P. E. T., Pedersen S., Dense analytic subspaces in fractal L2-spaces, J. Anal. Math., 1998, 75, 185-228
• [4] Dutkay D. E., Picioroaga G., Song M.-S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409, 1128-1139
• [5] Strichartz R. S., Mock fourier series and transforms associated with certain cantor measures, Journal d’Analyse Mathématique, 2000, 81, 209-238
• [6] Dovgoshey O., Martio O., Ryazanov V., Vuorinen M., The Cantor function, Expo. Math., 2006, 24, 1-37
• [7] Hsu E. P., A class of singular continuous functions, Elem. Math., 1992, 47, 169-172
• [8] Bogachev V. I., Measure theory. Vol. II, Springer-Verlag, Berlin, 2007
• [9] OEIS Foundation Inc. (2019), The On-Line Encyclopedia of Integer Sequences, http://oeis.org
• [10] Grabner P. J., Prodinger H., Asymptotic analysis of the moments of the Cantor distribution, Statist. Probab. Lett., 1996, 26, 243-248

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).