Beurling-Landau-Type theorems for non-uniform sampling in shift invariant spline spaces

Journal of Fourier Analysis and Applications

Volumn 6, Issue 1, 2000, Pages 93-103

  Aldroubi, Akram ^a   Gröchenig, Karlheinz ^a  

^a VANDERBILT UNIVERSITY   (United States)

Author keywords

Frames; Irregular sampling; Reproducing Kernel Hilbert spaces; Shift invariant spaces; Spline; Wiener amalgam spaces

Indexed keywords

EID: 52849111643     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf02510120     Document Type: Article

References (32)

1. Aldroubi, A. (1995). Portraits of frames, Proc. Am. Math. Soc., 123(6), 1661-1668.
2. p theory, Proc. Am. Math. Soc., 126(9), 2677-2686.
3. Aldroubi, A. and Unser, M. (1992). Families of wavelet transforms in connection with Shannon's sampling theory and the Gabor transform, in Wavelets: A Tutorial in Theory and Applications, C.K. Chui, Ed., Academic Press, 1992, 509-528.
4. Aldroubi, A. and Unser, M. (1994). Sampling procedure in function spaces and asymptotic equivalence with Shannon's sampling theory, Numer. Funct. Anal. and Optimiz., 15(1), 1-21.
5. Berenstein, C.A. and Patrick, E.V. (1990). Exact deconvolution for multiple convolution operators - an overview, plus performance characterizations for imaging sensors, in IEEE, 723-743. IEEE.
6. Beurling, A. (1989). In Carleson, L., Ed., A. Beurling. Collected Works. Vol. 2, Birkhäuser, Boston, 341-365.
7. Butzer, P.L. (1983). A survey of the Whittaker-Shannon sampling theorem and some of its extension, J. Math. Res. Exposition, 3(1), 185-212.
8. Chen, W., Itoh, S., and Shiki, J. (1998). Irregular sampling theorems for wavelet subspaces, IEEE Trans. Inform. Theory, 44, 1131-1142.
9. Chui, C.K., Ed. (1992). Wavelets: A Tutorial in Theory and Applications. Academic Press, San Diego, CA.
10. Duffin, R.J. and Schaeffer, A.C. (1952). A class of nonharmonic Fourier series, Trans. Am. Math. Soc., 72(2), 341-366.
11. Feichtinger, H.G. (1990). Generalized amalgams, with applications to Fourier transform, Can. J. Math., 42(3), 395-409.
12. Feichtinger, H.G. (1991). Wiener amalgams over euclidean spaces and some of their applications, in Jarosz, K., Ed., Proc. Conf. Function spaces, 136, of Lect. Notes in Math., Edwardsville, IL, April 1990, 123-137, Marcel-Dekker.
13. Feichtinger, H.G. and Gröchenig, H. (1992). Iterative reconstruction of multivariate band-limited functions from irregular sampling values, SIAM J. Math. Anal., 231, 244-261.
14. Feichtinger, H.G. and Gröchenig, K. (1993). Theory and practice of irregular sampling, in wavelets: Mathematics and applications, in Benedetto, J.J. and Frazier, M., Eds., Wavelets: Mathematics and Applications, CRC, Boca Raton, FL, 305-363.
15. q, Bull. Am. Math. Soc., 13(1), 1-21.
16. Gröchenig, K. (1992). Reconstruction algorithms in irregular sampling, Math. Comp., 59, 181-194.
17. Gröchenig, K. and Razafinjatovo, H. (1996). On Landau's necessary density conditions for sampling and interpolation of band-limited function, J. London Math. Soc., 54, 557-565.
18. Janssen, A. (1993). The Zak transform and sampling theorems for wavelet subspaces, IEEE Trans. Signal Process., 6, 39-49.
19. Jerri, A. (1977). The Shannon sampling theorem - its various extensions and applications: A tutorial review, Proc. IEEE, 65, 1565-1596.
20. Jia, R.Q. and Micchelli, C.A. (1991). Using the refinement equations for the construction of pre-wavelets. ii. powers of two, in Curves and Surfaces, Academic Press, Boston, MA, 209-246.
21. Landau, H. (1967). Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math., 117, 37-52.
22. Liu, Y. (1996). Irregular sampling for spline wavelet subspaces, IEEE Trans. Inform. Theory, 42, 623-627.
23. Liu, Y.M. and Walter, G.G. (1995). Irregular sampling in wavelet subspaces, J. Fourier Anal. Appl., 2(2), 181-189.
24. 2(ℝ), Trans. Am. Math. Soc., 315(1), 69-97.
25. Mallat, S. (1992). Personal communication, conf. oberwolfach, wavelets and applications.
26. Meyer, Y. (1990). Ondelettes et opérateurs. Hermann, Paris, France.
27. Schumaker, L. (1980). Spline Functions: Basic Theory. Wiley-Interscience, Boston.
28. Unser, M. and Aldroubi, A. (1994). A general sampling theory for non-ideal acquisition devices, IEEE Trans. on Signal Processing, 42(11), 2915-2925.
29. Walnut, D. (1996). Nonperiodic sampling of bandlimited functions on union of rectangular lattices, J. Fourier Anal. Appl., 2(5), 436-451.
30. Walter, G.G. (1992). A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory, part 2, 116(2), 881-884.
31. Xia, X.G. and Zhang, Z.Z. (1993). On sampling theorem, wavelets, and wavelet transforms, IEEE Trans. Signal Process., 41(12), 3524-3535.
32. Yao, K. (1967). Applications of reproducing kernel Hilbert spaces-bandlimited signal models, Information and Control, 11, 429-444.