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Notes on a theorem of Katznelson and Ornstein
Polynomial approximation of self-similar measures and the spectrum of the transfer operator
Institute of Mathematics, University of Greifswald, Germany |
We consider self-similar measures on $\mathbb{R}.$ The Hutchinson operator $H$ acts on measures and is the dual of the transfer operator $T$ which acts on continuous functions. We determine polynomial eigenfunctions of $T.$ As a consequence, we obtain eigenvalues of $H$ and natural polynomial approximations of the self-similar measure. Bernoulli convolutions are studied as an example.
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Ein Beitrag zur Theorie des Legendre'schen Polynoms, Acta Mathematica, 18 (1894), 155-159.
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J. E. Hutchinson,
Fractals and self-similarity, Indiana University Mathematics Journal, 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
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T. Kato,
Perturbation Theory for Linear Operators Classics in Mathematics, Springer Berlin Heidelberg, 1995. |
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P. Shmerkin,
On the exceptional set for absolute continuity of Bernoulli convolutions, Geometric and Functional Analysis, 24 (2014), 946-958.
doi: 10.1007/s00039-014-0285-4. |
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J. Shohat and J. Tamarkin,
The Problem of Moments Mathematical surveys and monographs, American Mathematical Society, 1943. |
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B. Solomyak, Notes on Bernoulli convolutions, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1, vol. 72 of Proceedings of Symposia in Pure Mathematics, AMS, 2004,207-230. |
[14] |
B. Solomyak, Y. Peres and W. Schlag,
Sixty years of Bernoulli convolutions, Progress in Probability, 46 (2000), 39-65.
|
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T. Tao,
An Epsilon of Room I Graduate Studies in Mathematics, American Mathematical Society, 2010.
doi: 10.1090/gsm/117. |
show all references
References:
[1] |
D. H. Bailey, J. M. Borwein, R. E. Crandall and M. G. Rose,
Expectations on fractal sets, Appl. Math. Comput., 220 (2013), 695-721.
doi: 10.1016/j.amc.2013.06.078. |
[2] |
K. Barański, Dimension of the graphs of the weierstrass-type functions, in Fractal Geometry and Stochastics V (eds. C. Bandt, K. Falconer and M. Zähle), vol. 70 of Progress in Probability, Springer International Publishing, 2015, 77-91.
doi: 10.1007/978-3-319-18660-3_5. |
[3] |
M. Barnsley and S. Demko,
Iterated function systems and the global construction of fractals, Proc. R. Soc. Lond., 399 (1985), 243-275.
doi: 10.1098/rspa.1985.0057. |
[4] |
B. Beckermann,
The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numerische Mathematik, 85 (2000), 553-577.
doi: 10.1007/PL00005392. |
[5] |
M.-D. Choi,
Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.
doi: 10.2307/2975779. |
[6] |
D. J. Driebe, The bernoulli map, in Fully Chaotic Maps and Broken Time Symmetry, vol. 4 of Nonlinear Phenomena and Complex Systems, Springer Netherlands, 1999, 19-43. |
[7] |
K. J. Falconer,
Fractal Geometry: Mathematical Foundations and Applications 3rd edition, J. Wiley & sons, 2014. |
[8] |
D. Hilbert,
Ein Beitrag zur Theorie des Legendre'schen Polynoms, Acta Mathematica, 18 (1894), 155-159.
doi: 10.1007/BF02418278. |
[9] |
J. E. Hutchinson,
Fractals and self-similarity, Indiana University Mathematics Journal, 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
[10] |
T. Kato,
Perturbation Theory for Linear Operators Classics in Mathematics, Springer Berlin Heidelberg, 1995. |
[11] |
P. Shmerkin,
On the exceptional set for absolute continuity of Bernoulli convolutions, Geometric and Functional Analysis, 24 (2014), 946-958.
doi: 10.1007/s00039-014-0285-4. |
[12] |
J. Shohat and J. Tamarkin,
The Problem of Moments Mathematical surveys and monographs, American Mathematical Society, 1943. |
[13] |
B. Solomyak, Notes on Bernoulli convolutions, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1, vol. 72 of Proceedings of Symposia in Pure Mathematics, AMS, 2004,207-230. |
[14] |
B. Solomyak, Y. Peres and W. Schlag,
Sixty years of Bernoulli convolutions, Progress in Probability, 46 (2000), 39-65.
|
[15] |
T. Tao,
An Epsilon of Room I Graduate Studies in Mathematics, American Mathematical Society, 2010.
doi: 10.1090/gsm/117. |




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