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Warianty tytułu
Języki publikacji
Abstrakty
We prove two sampling theorems for infinite (countable discrete) weighted graphs G; one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing G, and there are Hilbert spaces of functions on X that allow interpolation by sampling values of the functions restricted only on the vertices in G. We sample functions on X from their discrete values picked in the vertex-subset G. We prove two theorems that allow for such realistic ambient spaces X for a fixed graph G, and for interpolation kernels in function Hilbert spaces on X, sampling only from points in the subset of vertices in G. A continuum is often not apparent at the outset from the given graph G. We will solve this problem with the use of ideas from stochastic integration.
Czasopismo
Rocznik
Tom
Strony
209--236
Opis fizyczny
Bibliogr. 33 poz., rys., wykr.
Twórcy
autor
- The University of Iowa Department of Mathematics Iowa City, IA 52242-1419, USA, jorgen@math.uiowa.edu
Bibliografia
- [1] E. Acosta-Reyes, A. Aldroubi, I. Krishtal, On stability of sampling-reconstruction models, Adv. Comput. Math. 31 (2009) 1–3, 5–34.
- [2] A. Aldroubi, C. Cabrelli, C. Heil, K. Kornelson, U. Molter, Invariance of a shift--invariant space, J. Fourier Anal. Appl. 16 (2010) 1, 60–75.
- [3] A. Aldroubi, C. Cabrelli, C. Heil, K. Kornelson, U. Molter, Invariance of a shift--invariant space, J. Fourier Anal. Appl. 16 (2010) 1, 60–75.
- [4] A. Aldroubi, C. Cabrelli, U. Molter, Optimal non-linear models for sparsity and sampling,J. Fourier Anal. Appl. 14 (2008) 5–6, 793–812.
- [5] A. Aldroubi, C. Leonetti, Non-uniform sampling and reconstruction from sampling sets with unknown jitter, Sampl. Theory Signal Image Process. 7 (2008) 2, 187–195.
- [6] A. Aldroubi, C. Leonetti, Q. Sun, Error analysis of frame reconstruction from noisy samples, IEEE Trans. Signal Process. 56 (2008) 6, 2311–2325.
- [7] D. Alpay, D. Levanony, On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions, Potential Anal. 28 (2008) 2, 163–184.
- [8] D. Alpay, D. Levanony, Rational functions associated with the white noise space and related topics, Potential Anal. 29 (2008) 2, 195–220.
- [9] M.-J. Cantero, F. A. Grunbaum, L. Moral, L. Velazquez, Matrix-valued Szego polynomials and quantum random walks, Comm. Pure Appl. Math. 63 (2010) 4, 464–507.
- [10] I. Cho, P.E.T. Jorgensen, Applications of automata and graphs: labeling-operators in Hilbert space. I, Acta Appl. Math. 107 (2009) 1–3, 237–291.
- [11] I. Cho, P.E.T. Jorgensen, Applications of automata and graphs: labeling operators in Hilbert space. II, J. Math. Phys. 50 (6) (2009), 063511, 42.
- [12] I. Daubechies, R. DeVore, Approximating a bandlimited function using very coarsely quantized data: a family of stable sigma-delta modulators of arbitrary order, Ann. Of Math. (2) 158 (2003) 2, 679–710.
- [13] I. Daubechies, R. DeVore, M. Fornasier, C.S. Gunturk, Iteratively reweighted least squares minimization for sparse recovery, Comm. Pure Appl. Math. 63 (2010) 1, 1–38.
- [14] D. Dutkay, P.E.T. Jorgensen, Fourier series on fractals: a parallel with wavelet theory, [in:] Contemp. Math, Radon transforms, geometry, and wavelets, vol. 464, Amer. Math.Soc., Providence, RI, 2008.
- [15] Y. Ephraim,W.J.J. Roberts, An EM algorithm for Markov modulated Markov processes, IEEE Trans. Signal Process. 57 (2009) 2, 463–470.
- [16] F. Gokpinar, Y.A. Ozdemir, Generalization of inclusion probabilities in ranked set sampling, Hacet. J. Math. Stat. 39 (2010) 1, 89–95.
- [17] P.E.T. Jorgensen, Analysis of unbounded operators and random motion, J. Math. Phys.50 (2009) 11, 113503, 28.
- [18] P.E.T. Jorgensen, M.-S. Song, Analysis of fractals, image compression, entropy encoding, Karhunen-Loeve transforms, Acta Appl. Math. 108 (2009) 3, 489–508.
- [19] P.E.T. Jorgensen, M.-S. Song, An extension of Wiener integration with the use of operator theory, J. Math. Phys. 50 (2009) 10, 103502, 11.
- [20] P.E.T. Jorgensen, E.P.J. Pearse, Spectral Operator Theory of Electrical Resistance Networks, (2009), URL: http://arxiv.org/PS cache/arxiv/pdf/0806/0806.3881v4.pdf
- [21] Y. Katznelson, An introduction to harmonic analysis, Cambridge Mathematical Library, third edition, Cambridge University Press, Cambridge, 2004.
- [22] D. Kim, Y. Oshima, Some inequalities related to transience and recurrence of Markov processes and their applications, J. Theoret. Probab. 23 (2010) 1, 148–168.
- [23] B. Kummerer, Asymptotic behavior of quantum Markov processes, [in:] Infinite dimensional harmonic analysis IV, World Sci. Publ., Hackensack, NJ, 2009.
- [24] R.T. Powers, Resistance inequalities for the isotropic Heisenberg ferromagnet, J.Math. Phys. 17 (1976) 10, 1910–1918.
- [25] G.K. Rohde, A. Aldroubi, D.M. Healy Jr., Interpolation artifacts in sub-pixel image registration, IEEE Trans. Image Process. 18 (2009) 2, 333–345.
- [26] W. Rosenkrantz, Markov processes and applications: algorithms, networks, genome and finance [book review of mr2488952], SIAM Rev. 51 (2009) 4, 802–807.
- [27] C.E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423, 623–656.
- [28] C.E. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949), 10–21.
- [29] C.E. Shannon, W. Weaver, The Mathematical Theory of Communication, The University of Illinois Press, Urbana, IL., 1949.
- [30] B. Simon, Functional integration and quantum physics, vol. 86 of Pure and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979.
- [31] D. Ślęzak, Approximate Markov boundaries and Bayesian networks: rough set approach, [in:] Rough set theory and granular computing, vol. 125 of Stud. Fuzziness Soft Comput., Springer, Berlin, 2003.
- [32] E.A. Suess, B.E. Trumbo, Introduction to Probability Simulation and Gibbs Sampling with R, Springer, New York, 2010.
- [33] Q.-W. Xiao, Z.-W. Pan, Learning from non-identical sampling for classification, Adv. Comput. Math. 33 (2010) 1, 97–112.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHT-0005-0004