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Applications of optimally local interpolation
to interpolatory approximants
and compactly supported wavelets

Authors: Charles K. Chui and Johan M. De Villiers
Journal: Math. Comp. 65 (1996), 99-114
MSC (1991): Primary 41A05, 41A15
MathSciNet review: 1322886
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Abstract: The objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary (nonuniform) knots. Both construction procedures are based on certain ``optimally local'' interpolatory fundamental spline functions which are not required to possess any approximation property.

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  • 1 C. de Boor, A practical guide to splines, Appl. Math. Sci. #27, Springer-Verlag, New York, 1978, MR 80a:65027.
  • 2 M. Buhmann and C. A. Micchelli, Spline pre-wavelets for non-uniform knots, Numer. Math. 61 (1992), 455--474, MR 93g:41021.
  • 3 G. Chen, C. K. Chui, and M. J. Lai, Construction of real-time spline quasi-interpolation schemes, Approx. Theory Appl. 4 (1988), 61--75, MR 90d:41015.
  • 4 C. K. Chui, Construction and applications of interpolation formulas, Multivariate Approximation and Interpolation (W. Haussmann and K. Jetter, eds.), Internat. Ser. Numer. Math., vol. 94, Birkhäuser Verlag, Basel, 1990, pp. (11--23), MR 92c:41002.
  • 5 ------, An introduction to wavelets, Academic Press, Boston, 1992, MR 93f:42055.
  • 6 C. K. Chui and H. Diamond, A general framework for local interpolation, Numer. Math. 58 (1991), 569--581, MR 92b:41005.
  • 7 C. K. Chui and E. Quak, Wavelets on a bounded interval, Numerical Methods in Approximation Theory, vol. 9 (D. Braess and L.L. Schumaker, eds.), Internat. Ser. Numer. Math., vol. 105, Birkhäuser Verlag, Basel, 1992, pp. (53--75), MR 95b:42027.
  • 8 C. K. Chui and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), 785--793, MR 92b:41019.
  • 9 ------, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992), 903--915, MR 92f:41020.
  • 10 ------, An analysis of cardinal spline-wavelets, J. Approx. Theory 72 (1993), 54--68, MR 94f:42041.
  • 11 W. Dahmen, T. N. T. Goodman, and C. A. Micchelli, Compactly supported fundamental functions for spline interpolation, Numer. Math. 52 (1988), 639--664, MR 89i:65011.
  • 12 ------, Local spline interpolation schemes in one and several variables, Approximation and Optimization, Lecture Notes in Math., vol. 1354, Springer, Berlin, 1989, pp. (11--24), MR 90b:41017.
  • 13 I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Math. #61, SIAM, Philadelphia, PA, 1992, MR 93e:42045.
  • 14 J. M. De Villiers, A convergence result in nodal spline interpolation, J. Approx. Theory 74 (1993), 266--279, MR 94i:41013.
  • 15 J. M. De Villiers and C. H. Rohwer, Optimal local spline interpolants, J. Comput. Appl. Math. 18 (1987), 107--119, MR 88i:41017.
  • 16 ------, A nodal spline generalization of the Lagrange interpolant, Progress in Approximation Theory (P. Nevai and A. Pinkus, eds.), Academic Press, San Diego, 1991, pp. (201--212), MR 92h:41004.
  • 17 ------, Sharp bounds for the Lebesgue constant in quadratic nodal spline interpolation, Approximation and Computation (R.V.M. Zahar, ed.), Internat. Ser. Numer. Math. #119, Birkhäuser Verlag, Basel, 1994, pp. (157--167).
  • 18 G. H. Golub and C. F. Van Loan, Matrix computations, Johns Hopkins Univ. Press, Baltimore, MD, 1985, MR 85h:65063.
  • 19 T. N. T. Goodman, S. L. Lee, and W. S. Tang, Wavelets in wandering spaces, Trans. Amer. Math. Soc. 338 (1993), 639--654, MR 93j:42017.
  • 20 S. Karlin, Total positivity, Stanford Univ. Press, Stanford, CA, 1968, MR 37:5667.
  • 21 T. Lyche and K. Mørken, Spline-wavelets of minimum support, Numerical Methods in Approximation Theory, Vol. 9 (D. Braess and L.L. Schumaker, eds.), Internat. Ser. Numer. Math., Vol. 105, Birkhäuser Verlag, Basel, 1992, pp. (177--192), MR 95c:41026.
  • 22 L. L. Schumaker, Spline functions: basic theory, Wiley-Interscience, New York, 1981, MR 82j:41001.

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Additional Information

Charles K. Chui
Affiliation: Center for Approximation Theory, Texas A&M University, College Station, Texas 77843

Johan M. De Villiers
Affiliation: Center for Approximation Theory, Texas A&M University, College Station, Texas 77843

Received by editor(s): December 17, 1993
Additional Notes: Research of the first author was supported by NSF Grant DMS 92-06928 and ARO Contract DAAH 03-93-G-0047.
Article copyright: © Copyright 1996 American Mathematical Society

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