Approximation error for quasi-interpolators and (multi-)wavelet expansions

Applied and Computational Harmonic Analysis

Volumn 6, Issue 2, 1999, Pages 219-251

  Blu, Thierry ^a   Unser, Michael ^a  

^a EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000751923     PISSN: 10635203     EISSN: None     Source Type: Journal    
DOI: 10.1006/acha.1998.0249     Document Type: Article

References (51)

1. A. Aldroubi, Oblique and hierarchical multiwavelet bases, Appl. Comput. Harmon. Anal. 4 (1997), 231-263.
2. A. Aldroubi, Oblique projections in atomic spaces, Proc. Amer. Math. Soc. 124 (1996), 2051-2060.
3. A. Aldroubi and M. Unser, Families of multiresolution and wavelet spaces with optimal properties, Numer. Funct. Anal. Opt. 14 (1993), 417-446.
4. A. Aldroubi and M. Unser, Sampling procedures in function spaces and asymptotic equivalence with Shannon's sampling theory, Numer. Funct. Anal. Opt. 15 (1994), 1-21.
5. M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, Image coding using wavelet transform, IEEE Trans. Image Process. 1 (1992), 205-220.
6. G. Battle, A block spin construction of ondelettes. I. Lemarié functions, Comm. Math. Phys. 110 (1987), 601-615.
7. T. Blu, Iterated filter banks with rational rate changes - Connection with discrete wavelet transforms, IEEE Trans. Signal Process. 41 (1993), 3232-3244.
8. T. Blu and M. Unser, Quantitative Fourier analysis of approximation techniques, IEEE Trans. Signal Process., submitted.
9. H. Burchard and J. Lei, Coordinate order of approximation by functional-based approximation operators, J. Approx. Theory 82 (1995), 240-256.
10. E. Cheney and W. Light, Quasi-interpolation with basis functions having non-compact support, Constr. Approx. 8 (1992), 35-48.
11. C. Chui and H. Diamond, A characterization of multivariate quasi-interpolation formulas and applications, Numer. Math. 57 (1990), 105-121.
12. C. Chui and J. Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992), 903-915.
13. A. Cohen and I. Daubechies, Orthonormal bases of compactly supported wavelets. III. Better frequency resolution, SIAM J. Math. Anal. 24 (1993), 520-527.
14. A. Cohen, I. Daubechies, and J. Feauveau, Biorthogonal basis of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485-560.
15. A. Cohen, I. Daubechies, and G. Plonka, Regularity of refinable function vectors, J. Fourier Anal. Appl. (1997), 295-324.
16. W. Dahmen and C. Micchelli, On the approximation order from certain multivariate spline spaces, J. Austral. Math. Soc. Ser. B 26 (1984), 233-246.
17. W. Dahmen and C. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293-328.
18. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996.
19. I. Daubechies, "Ten Lectures on Wavelets," SIAM, Philadelphia, PA, 1992.
20. C. de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx. 3 (1987), 199-208.
21. C. de Boor, Quasi-interpolation and approximation power of multivariate splines, in "Computation of Curves and Surfaces" (W. D. et al., Eds.), pp. 313-345, Kluwer, Dordrecht, 1990.
22. d), Trans. Amer. Math. Soc. 341 (1994), 787-806.
23. d), J. Funct. Anal. 119 (1994).
24. C. de Boor and G. Fix, Spline approximation by quasi-interpolants, J. Approx. Theory 8 (1973), 19-45.
25. C. de Boor and R. Jia, Controlled approximation and a characterization of the local approximation order, Proc. Amer. Math. Soc. 95 (1985), 547-553.
26. T. Goodman and S. Lee, Wavelets of multiplicity r, Trans. Amer. Math. Soc. 342 (1993), 307-324.
27. C. Heil, G. Strang, and V. Strela, Approximation by translates of refinable functions, Numer. Math 73 (1996), 75-94.
28. L. Hervé, Multi-resolution analysis of multiplicity d. Application to dyadic interpolation, Appl. Comput. Harmon. Anal. 1 (1994), 299-315.
29. K. Jetter and D. Zhou, Order of linear approximation from shift invariant spaces, Constr. Approx. 11 (1995), 423-438.
30. R. Jia, A counterexample to a result concerning controlled approximation, Trans. Amer. Math. Soc. 97 (1986), 647-654.
31. R. Jia, Shift-invariant spaces on the real line, Proc. Amer. Math. Soc. 125 (1997), 785-793.
32. R. Jia and J. Lei, Approximation by multiinteger translates of functions having global support, J. Approx. Theory 72 (1993), 2-23.
33. p-approximation by certain projection operators, J. Math. Anal. Appl. 185 (1994), 1-14.
34. P.-G. Lemarié, Ondelettes à localisation exponentielle, J. Math. Pures Appl. 67 (1988), 227-236. [In French]
35. S. Mallat, A theory for multiresolution signal decomposition: The wavelet decomposition, IEEE Trans. Pattern Anal. Mach. Intell. 11 (1989), 674-693.
36. Y. Meyer, "Ondelettes," Hermann, Pans, 1990. [In French]
37. G. Plonka, Approximation order provided by refinable function vectors, Constr. Approx. 13 (1997), 221-244.
38. O. Rioul, On the choice of wavelet filters for still image compression, in "Proc. IEEE Int. Conf. Acoust, Speech, Sig. Proc.," Vol. V, pp. 550-553, Minneapolis, MN, Apr. 1993.
39. I. Schoenberg, Contribution to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946), 45-99, 112-141.
40. L. Schwartz, "Théorie des Distributions," Hermann, Paris, 1966. [In French]
41. C.Shannon, Communication in the presence of noise, in "Proc. IRE," Vol. 37, pp. 10-21, Jan. 1949.
42. G. Strang and G. Fix, A Fourier analysis of the finite element variational method, in "Constructive Aspect of Functional Analysis" (Cremonese, Ed.), pp. 796-830, Rome, 1971.
43. G. Strang and T. Nguyen, £Wavelets and Filter Banks,£ Wellesley-Cambridge Press, Cambridge, MA, 1996.
44. W. Sweldens and R. Piessens, Quadrature formulas and asymptotic error expansions for wavelet approximations of smooth functions, SIAM J. Math. Anal. 31 (1994), 1240-1264.
45. M. Unser, Approximation power of biorthogonal wavelet expansions, IEEE Trans. Signal Process. 44 (1996), 519-527.
46. M. Unser, Quasi-orthogonality and quasi-projections, Appl. Comput. Harmon. Anal. 3 (1996), 201-214.
47. M. Unser, A. Aldroubi, and M. Eden, On the asymptotic convergence of B-spline wavelets to Gabor functions, IEEE Trans. Inform. Theory 38 (1992), 864-872.
48. M. Unser and I. Daubechies, On the approximation power of convolution-based least squares versus interpolation, IEEE Trans. Signal Process. 45, (1997), 1697-1711.
49. M. Vetterli and C. Herley, Wavelets and filter banks: Theory and design, IEEE Trans. Signal Process. 40 (1992), 2207-2232.
50. E. Whittaker, The Fourier theory of the cardinal functions, Proc. Math. Soc. Edinburgh 1 (1929), 169-176.
51. X. Xia and B. Suter, Vector-valued wavelets and vector filter banks, IEEE Trans. Signal Process. 44 (1996), 508-518.