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Locally finite dimensional shift-invariant spaces in $\mathbf{R}^d$

Authors: Akram Aldroubi and Qiyu Sun
Journal: Proc. Amer. Math. Soc. 130 (2002), 2641-2654
MSC (2000): Primary 42C40, 46A35, 46E15
Published electronically: February 12, 2002
MathSciNet review: 1900872
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Abstract: We prove that a locally finite dimensional shift-invariant linear space of distributions must be a linear subspace of some shift-invariant space generated by finitely many compactly supported distributions. If the locally finite dimensional shift-invariant space is a subspace of the Hölder continuous space $C^\alpha$ or the fractional Sobolev space $L^{p, \gamma}$, then the superspace can be chosen to be $C^\alpha$ or $L^{p, \gamma}$, respectively.

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  • 1. A. Aldroubi and M. Unser. Sampling procedures in function spaces and asymptotic equivalence with Shannon's sampling theory, Numer. Funct. Anal. Optimiz., 15(1)(1994), 1-21. MR 95a:94002
  • 2. A. Aldroubi, Q. Sun and W.-S. Tang, $p$-frames and shift invariant subspaces of $L^p$, J. Fourier Anal. Appl., 7(1)(2001), 1-21. CMP 2001:08
  • 3. A. Ben-Artzi and A. Ron, On the integer translates of a compactly supported function: dual bases and linear projectors, SIAM J. Math. Anal., 21(1990), 1550-1562. MR 91j:41009
  • 4. C. de Boor and R. Devore, Partitions of unity and approximation, Proc. Amer. Math. Soc., 93(1985), 705-709. MR 86f:41003
  • 5. C. de Boor, R. Devore and A. Ron, The structure of finitely generated shift-invariant spaces in $L_2(\mathbf{R}^d)$, J. Funct. Anal., 119(1994), 37-78. MR 95g:46050
  • 6. C. Cabrelli, C. Heil and U. Molter, Accuracy of several multidimensional refinable distributions, J. Fourier Anal. Appl. 6(2000), 483-502. MR 2001g:65179
  • 7. W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52(1983), 217-234. MR 85e:41033
  • 8. T. N. T. Goodman, S. L. Lee and W. S. Tang, Wavelets in wandering subspaces, Trans. Amer. Math. Soc., 338(1993), 639-654. MR 93j:42017
  • 9. D. Hardin and T. Hogan, Refinable subspaces of a refinable space, Proc. Amer. Math. Soc., 128(2000), 1941-1950. MR 2000m:42027
  • 10. T. Hogan, Stability and independence of the shifts of finitely many refinable functions, J. Fourier Anal. Appl., 3(1997), 758-774. MR 99c:42061
  • 11. R.-Q. Jia, Shift-invariant spaces and linear operator equations, Israel Math. J., 103(1998), 259-288. MR 99d:41016
  • 12. R. Q. Jia, Stability of the shifts of a finite number of functions, J. Approx. Th., 95(1998), 194-202. MR 99h:42040
  • 13. R.-Q. Jia and C. A. Micchelli, Using the refinement equation for the construction of pre-wavelets II: power of two, In ``Curve and Surface" (P. J. Laurent, A. Le Méhaute and L. L. Schumaker eds.), Academic Press, New York 1991, pp. 209-246. MR 93e:65024
  • 14. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, New York 1998. MR 99m:94012
  • 15. A. Ron, Introduction to shift-invariant spaces I: linear independence, Multivariate Approximation and Applications (N. Dyn et al., eds.), Cambridge Univ. Press, Cambridge, 2001, pp. 112-151.
  • 16. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. MR 44:7280
  • 17. Q. Sun, Convergence of cascade algorithms and smoothness of refinable distributions, Preprint 1999.
  • 18. H. Triebel, Theory of Function Spaces II, Monographs in Math. No. 84, Birkhäuser, 1992. MR 93f:46029
  • 19. J. M. Whittaker, Interpolatory Function Theory, Cambridge University Press, London, 1935. MR 32:2798 (reprint)
  • 20. K. Zhao, Global linear independence and finitely supported dual basis, SIAM J. Math. Anal., 23(1992), 1352-1355. MR 93h:41034

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Additional Information

Akram Aldroubi
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennnessee 37240

Qiyu Sun
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

Keywords: Fractional Sobolev spaces, H\"older continuous, distributions
Received by editor(s): October 27, 2000
Received by editor(s) in revised form: April 2, 2001
Published electronically: February 12, 2002
Additional Notes: The first author’s research was supported in part by NSF grant DMS-9805483.
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

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