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Transactions of the American Mathematical Society

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Random variable dilation equation and multidimensional prescale functions


Authors: Julie Belock and Vladimir Dobric
Journal: Trans. Amer. Math. Soc. 353 (2001), 4779-4800
MSC (2000): Primary 60A10, 60G50; Secondary 42C40, 42C15
DOI: https://doi.org/10.1090/S0002-9947-01-02833-1
Published electronically: June 21, 2001
MathSciNet review: 1852082
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Abstract:

A random variable $Z$ satisfying the random variable dilation equation $MZ \overset{d}{=}Z+G$, where $G$ is a discrete random variable independent of $Z $ with values in a lattice $\Gamma \subset $ $\mathbf{R}^{d}$ and weights $\left\{ c_{k}\right\} _{k\in \Gamma }$ and $M$ is an expanding and $\Gamma $-preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density $\varphi $ which will satisfy a dilation equation

\begin{displaymath}\varphi \left( x\right) =\left\vert \det M\right\vert \sum_{k\in \Gamma} c_{k}\varphi \left( Mx-k\right) \text{.} \end{displaymath}

We have obtained necessary and sufficient conditions for the existence of the density $\varphi $ and a simple sufficient condition for $\varphi $'s existence in terms of the weights $\left\{ c_{k}\right\} _{k\in \Gamma }.$Wavelets in $\mathbf{R}^{d}$ can be generated in several ways. One is through a multiresolution analysis of $L^{2}\left( \mathbf{R}^{d}\right) $ generated by a compactly supported prescale function $\varphi $. The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of $\varphi $ allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when $\varphi $ is a prescale function.


References [Enhancements On Off] (What's this?)

  • 1. C. Bandt, Self-similar sets. V. Integer matrices and fractal tilings of R$^{n},$ Proc. Amer. Math. Soc., 112, (2) (1991), pp. 549-562. MR 92d:58093
  • 2. D. Colella and C. Heil, Dilation equations and the smoothness of compactly supported wavelets in Wavelets: Mathematics and Applications (J. Benedetto and M. Frazier, eds.), CRC Press, Boca Raton, FL, 1994, pp. 163-201. MR 94j:42049
  • 3. I. Daubechies, Ten Lectures on Wavelets, CMBS Regional Conference Series in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA, 1992. MR 93e:42045
  • 4. V. Dobric, R.F. Gundy, P. Hitczenko, Characterizations of orthonormal scale functions: a probabilistic approach, J. Geom. Anal. 10 (2000), pp. 417-434. CMP 2001:04
  • 5. K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases and self-similar tilings of $\mathbf{R}^{n},$ IEEE Trans. Inform. Theory, 38 (2) (1992), pp. 556-567. MR 93i:42001
  • 6. R. Gundy and C. Zhang, ``Dilation equations,'' Lehigh University Probability Seminar, Fall, 1994.
  • 7. J. Hoffmann-Jørgensen, Measures which agree on balls , Math. Scand., 37 (1975), pp. 319-326. MR 53:13509
  • 8. B. Jessen and A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc., 38 (1935), pp. 44-88. CMP 95:18
  • 9. Q. Jiang, Multivariate matrix refinable functions with arbitrary matrix dilation, Trans. Amer. Math. Soc., 351 (6) (1999), pp. 2407-2438. MR 99i:42047
  • 10. J. Lagarias and Y. Wang, Self-Affine Tiles in $ \mathbf{R}^{n}$, Adv. Math., 121 (1996), pp. 21-49. MR 97d:52034
  • 11. W. Lawton, S. L. Lee, Z. Shen, Stability and orthonormality of multivariate refinable functions, SIAM J. Math. Anal., 28 (4) (1997), pp. 999-1014. MR 98d:41027
  • 12. P. G. Lemarie, Fonctions a support compact dans les analyses multi-resolutions, Rev. Mat. Iberoamericana, 7 (2) (1991), pp. 157-182. MR 93b:42046
  • 13. C. A., Rogers, Analytic Sets, Academic Press, New York, 1980.
  • 14. G. Strang, Wavelet transforms versus Fourier transforms, Bull. Amer. Math. Soc., 28 (2) (1993), pp. 288-305. MR 94b:42017

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Additional Information

Julie Belock
Affiliation: Department of Mathematics, West Chester University of Pennsylvania, West Chester, Pennsylvania 19383
Address at time of publication: Department of Mathematics, Salem State College, Salem, Massachusetts 01970
Email: jbelock@salemstate.edu

Vladimir Dobric
Affiliation: Department of Mathematics, Lehigh University, 14 Packer Avenue, Bethlehem, Pennsylvania 18015
Email: vd00@lehigh.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02833-1
Keywords: Dilation equations, tilings, wavelets
Received by editor(s): January 10, 2000
Received by editor(s) in revised form: January 8, 2001
Published electronically: June 21, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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