Eigenvalues of scaling operators and a characterization of $B$-splines
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- by Xiaojie Gao, S. L. Lee and Qiyu Sun PDF
- Proc. Amer. Math. Soc. 134 (2006), 1051-1057 Request permission
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.References
- Ola Bratteli and Palle Jorgensen, Wavelets through a looking glass, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2002. The world of the spectrum. MR 1913212, DOI 10.1007/978-0-8176-8144-9
- C. A. Cabrelli, S. B. Heineken and U. M. Molter, Local Bases for refinable spaces, preprint.
- Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, DOI 10.1090/memo/0453
- Charles K. Chui, An introduction to wavelets, Wavelet Analysis and its Applications, vol. 1, Academic Press, Inc., Boston, MA, 1992. MR 1150048
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
- Ingrid Daubechies and Jeffrey C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), no. 5, 1388–1410. MR 1112515, DOI 10.1137/0522089
- Timo Eirola, Sobolev characterization of solutions of dilation equations, SIAM J. Math. Anal. 23 (1992), no. 4, 1015–1030. MR 1166573, DOI 10.1137/0523058
- Xiaojie Gao, S. L. Lee, and Qiyu Sun, Spectrum of transition, subdivision and multiscale operators, Wavelet analysis (Hong Kong, 2001) Ser. Anal., vol. 1, World Sci. Publ., River Edge, NJ, 2002, pp. 123–138. MR 1941608
- Rong Qing Jia, Subdivision schemes in $L_p$ spaces, Adv. Comput. Math. 3 (1995), no. 4, 309–341. MR 1339166, DOI 10.1007/BF03028366
- Rong-Qing Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998), no. 222, 647–665. MR 1451324, DOI 10.1090/S0025-5718-98-00925-9
- Rong-Qing Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4089–4112. MR 1473444, DOI 10.1090/S0002-9947-99-02185-6
- Wayne M. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys. 32 (1991), no. 1, 57–61. MR 1083085, DOI 10.1063/1.529093
- Wayne Lawton, S. L. Lee, and Zuowei Shen, Characterization of compactly supported refinable splines, Adv. Comput. Math. 3 (1995), no. 1-2, 137–145. MR 1314906, DOI 10.1007/BF03028364
- Amos Ron and Zuowei Shen, The Sobolev regularity of refinable functions, J. Approx. Theory 106 (2000), no. 2, 185–225. MR 1788272, DOI 10.1006/jath.2000.3482
- Qiyu Sun and Zeyin Zhang, A characterization of compactly supported both $m$ and $n$ refinable distributions, J. Approx. Theory 99 (1999), no. 1, 198–216. MR 1696549, DOI 10.1006/jath.1998.3303
- M. Unser, Splines: a perfect fit for signal and image processing, IEEE Signal and Image Proc. Magazine, 16(1999), no. 6., 22–38.
- Lars F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal. 25 (1994), no. 5, 1433–1460. MR 1289147, DOI 10.1137/S0036141092228179
- 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.
- Ding-Xuan Zhou, Spectra of subdivision operators, Proc. Amer. Math. Soc. 129 (2001), no. 1, 191–202. MR 1784023, DOI 10.1090/S0002-9939-00-05727-0
- Ding-Xuan Zhou, Two-scale homogeneous functions in wavelet analysis, J. Fourier Anal. Appl. 8 (2002), no. 6, 565–580. MR 1932746, DOI 10.1007/s00041-002-0027-0
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
- Received by editor(s): October 25, 2004
- Published electronically: July 21, 2005
- Communicated by: David R. Larson
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2196038