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Eigenvalues of scaling operators and a characterization of $B$-splines


Authors: Xiaojie Gao, S. L. Lee and Qiyu Sun
Journal: Proc. Amer. Math. Soc. 134 (2006), 1051-1057
MSC (2000): Primary 41A15, 41A99, 42C40, 65T60
DOI: https://doi.org/10.1090/S0002-9939-05-08092-5
Published electronically: July 21, 2005
MathSciNet review: 2196038
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Abstract: A finitely supported sequence $a$ that sums to $2$ defines a scaling operator $ T_a f = \sum_{k\in \mathbb Z} a(k)f(2 \cdot -k)$ on functions $f,$ a transition operator $S_a v = \sum_{k\in \mathbb Z} a(k) (2 \cdot -k)$ on sequences $v,$ and a unique compactly supported scaling function $\phi$ that satisfies $\phi = T_a \phi$normalized with $\widehat \phi (0) = 1.$ It is shown that the eigenvalues of $T_a$ on the space of compactly supported square-integrable functions are a subset of the nonzero eigenvalues of the transition operator $S_a$ on the space of finitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function $\phi$is a uniform $B$-spline.


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  • 1. O. Bratteli and P. Jorgensen, Wavelets through a looking glass, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1913212 (2003i:42001)
  • 2. C. A. Cabrelli, S. B. Heineken and U. M. Molter, Local Bases for refinable spaces, preprint.
  • 3. A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision, Memoir Amer. Math. Soc., 93(1991), 1-186. MR 1079033 (92h:65017)
  • 4. C. K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992. MR 1150048 (93f:42055)
  • 5. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992. MR 1162107 (93e:42045)
  • 6. I. Daubechies and J. Lagarias, Two-scale difference equation I: existence and global regularity of solutions, SIAM J. Math. Anal., 22(1991), 1388-1410. MR 1112515 (92d:39001)
  • 7. T. Eirola, Sobolev characterization of solutions of dilation equation, SIAM J. Math. Anal., 23(1992), 1015-1030. MR 1166573 (93f:42056)
  • 8. X. Gao, S. L. Lee and Q. Sun, Spectrum of Transition, Subdivision and Multiscale operators, Wavelet Analysis (Hong Kong 2001) (edited by D. Zhou), World Science Publishing, 2002, 123-138. MR 1941608 (2003j:42034)
  • 9. R. Q. Jia, Subdivision schemes in $L^p$ spaces, Adv. Comput. Math., 3(1995), 309-341. MR 1339166 (96d:65028)
  • 10. R. Q. Jia, Approximation properties of multivariate wavelets, Math. Comp., 67(1998), 647-665. MR 1451324 (98g:41020)
  • 11. R. Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc., 351(1999), 4089-4112. MR 1473444 (99m:42050)
  • 12. W. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J Math. Physics, 32(1991), 57-61. MR 1083085 (91m:81100)
  • 13. W. Lawton, S. L. Lee and Z. Shen, Characterization of compactly supported refinable splines, Adv. Comput. Math., 3(1995), 137-145. MR 1314906 (95m:41020)
  • 14. A. Ron and Z. Shen, The Sobolev regularity of refinable functions, J. Approx. Theory, 106(2000), 185-225. MR 1788272 (2001j:42034)
  • 15. Q. Sun and Z. Zhang, A characterization of compactly supported both $m$ and $n$ refinable distributions, J. Approx. Theory, 99(1999), 198-216. MR 1696549 (2001e:42051)
  • 16. M. Unser, Splines: a perfect fit for signal and image processing, IEEE Signal and Image Proc. Magazine, 16(1999), no. 6., 22-38.
  • 17. L. F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal., 25(1994), 1433-1460. MR 1289147 (96f:39009)
  • 18. Y. P. Wang and S. L. Lee, Scale-space derived from $B$-spline, IEEE Trans. on Pattern Analysis and Machine Intelligence, 20(1998), 1040-1055.
  • 19. D. Zhou, Spectra of subdivision operators, Proc. Amer. Math. Soc. 129(2001), no. 1, 191-202. MR 1784023 (2001h:47049)
  • 20. D. Zhou, Two-scale homogeneous functions in wavelet analysis, J. Fourier Anal. Appl., 8(2002), no. 6, 565-580. MR 1932746 (2003i:42060)

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Additional Information

Xiaojie Gao
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matgxj@nus.edu.sg

S. L. Lee
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matleesl@nus.edu.sg

Qiyu Sun
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matsungy@nus.edu.sg

DOI: https://doi.org/10.1090/S0002-9939-05-08092-5
Keywords: Scaling operators, transition operators, eigenvalues, uniform $B$-splines
Received by editor(s): October 25, 2004
Published electronically: July 21, 2005
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society

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