Fourier spectra of measures associated with algorithmically random Brownian motion
In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have,...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Logical Methods in Computer Science e.V.
2014-09-01
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| Series: | Logical Methods in Computer Science |
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| Online Access: | https://lmcs.episciences.org/819/pdf |
| _version_ | 1797268658489131008 |
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| author | Willem Louw Fouché Safari Mukeru George Davie |
| author_facet | Willem Louw Fouché Safari Mukeru George Davie |
| author_sort | Willem Louw Fouché |
| collection | DOAJ |
| description | In this paper we study the behaviour at infinity of the Fourier transform of
Radon measures supported by the images of fractal sets under an algorithmically
random Brownian motion. We show that, under some computability conditions on
these sets, the Fourier transform of the associated measures have, relative to
the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity.
The argument relies heavily on a direct characterisation, due to Asarin and
Pokrovskii, of algorithmically random Brownian motion in terms of the prefix
free Kolmogorov complexity of finite binary sequences. The study also
necessitates a closer look at the potential theory over fractals from a
computable point of view. |
| first_indexed | 2024-04-25T01:35:59Z |
| format | Article |
| id | doaj.art-bfb8f0c9198348468ab2784caa2a7494 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:35:59Z |
| publishDate | 2014-09-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-bfb8f0c9198348468ab2784caa2a74942024-03-08T09:37:13ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742014-09-01Volume 10, Issue 310.2168/LMCS-10(3:20)2014819Fourier spectra of measures associated with algorithmically random Brownian motionWillem Louw FouchéSafari MukeruGeorge DavieIn this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.https://lmcs.episciences.org/819/pdfcomputer science - computational complexity |
| spellingShingle | Willem Louw Fouché Safari Mukeru George Davie Fourier spectra of measures associated with algorithmically random Brownian motion Logical Methods in Computer Science computer science - computational complexity |
| title | Fourier spectra of measures associated with algorithmically random Brownian motion |
| title_full | Fourier spectra of measures associated with algorithmically random Brownian motion |
| title_fullStr | Fourier spectra of measures associated with algorithmically random Brownian motion |
| title_full_unstemmed | Fourier spectra of measures associated with algorithmically random Brownian motion |
| title_short | Fourier spectra of measures associated with algorithmically random Brownian motion |
| title_sort | fourier spectra of measures associated with algorithmically random brownian motion |
| topic | computer science - computational complexity |
| url | https://lmcs.episciences.org/819/pdf |
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