Abstract
Under the appropriate definition of sampling density Dϕ, a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if Dϕ≥1. This result is similar to Landau's result for the Paley-Wiener space B 1/2 . If the shift invariant space consists of polynomial splines, then we show that Dϕ<1 is sufficient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B 1/2 .
Similar content being viewed by others
References
Aldroubi, A. (1995). Portraits of frames,Proc. Am. Math. Soc.,123(6), 1661–1668.
Aldroubi, A. and Feichtinger, H. (1998). Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: theL p theory,Proc. Am. Math. Soc.,126(9), 2677–2686.
Aldroubi, A. and Unser, M. (1992). Families of wavelet transforms in connection with Shannon's sampling theory and the Gabor transform, inWavelets: A Tutorial in Theory and Applications, C.K. Chui, Ed., Academic Press, 1992, 509–528.
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.
Berenstein, C.A. and Patrick, E.V. (1990). Exact deconvolution for multiple convolution operators—an overview, plus performance characterizations for imaging sensors, inIEEE, 723–743. IEEE.
Beurling, A. (1989). In Carleson, L., Ed.,A. Beurling. Collected Works. Vol. 2, Birkhäuser, Boston, 341–365.
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.
Chen, W., Itoh, S., and Shiki, J. (1998). Irregular sampling theorems for wavelet subspaces,IEEE Trans. Inform. Theory,44, 1131–1142.
Chui, C.K., Ed. (1992).Wavelets: A Tutorial in Theory and Applications. Academic Press, San Diego, CA.
Duffin, R.J. and Schaeffer, A.C. (1952). A class of nonharmonic Fourier series,Trans. Am. Math. Soc.,72(2), 341–366.
Feichtinger, H.G. (1990). Generalized amalgams, with applications to Fourier transform,Can. J. Math.,42 (3), 395–409.
Feichtinger, H.G. (1991). Wiener amalgams over euclidean spaces and some of their applications, in Jarosz, K., Ed.,Proc. Conf. Function spaces,136, ofLect. Notes in Math., Edwardsville, IL, April 1990, 123–137, Marcel-Dekker.
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.
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.
Fournier, J.J. and Stewart, J. (1985). Amalgams ofl p andl p.Bull. Am. Math. Soc.,13(1), 1–21.
Gröchenig, K. (1992). Reconstruction algorithms in irregular sampling,Math. Comp.,59, 181–194.
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.
Janssen, A. (1993). The Zak transform and sampling theorems for wavelet subspaces,IEEE Trans. Signal Process.,6, 39–49.
Jerri, A. (1977). The Shannon sampling theorem—its various extensions and applications: A tutorial review,Proc. IEEE,65, 1565–1596.
Jia, R.Q. and Micchelli, C.A. (1991). Using the refinement equations for the construction of pre-wavelets. ii. powers of two, inCurves and Surfaces, Academic Press, Boston, MA, 209–246.
Landau, H. (1967). Necessary density conditions for sampling and interpolation of certain entire functions,Acta Math.,117, 37–52.
Liu, Y. (1996). Irregular sampling for spline wavelet subspaces,IEEE Trans. Inform. Theory,42, 623–627.
Liu, Y.M. and Walter, G.G. (1995). Irregular sampling in wavelet subspaces,J. Fourier Anal. Appl.,2(2), 181–189.
Mallat, S. (1989). Multiresolution approximations and wavelet orthonormal bases of L2ℝ,Trans. Am. Math. Soc.,315(1), 69–97.
Mallat, S. (1992). Personal communication, conf. oberwolfach, wavelets and applications.
Meyer, Y. (1990).Ondelettes et opérateurs, Hermann, Paris, France.
Schumaker, L. (1980).Spline Functions: Basic Theory. Wiley-Interscience, Boston.
Unser, M. and Aldroubi, A. (1994). A general sampling theory for non-ideal acquisition devices,IEEE Trans. on Signal Processing,42(11), 2915–2925.
Walnut, D. (1996). Nonperiodic sampling of bandlimited functions on union of rectangular lattices,J. Fourier Anal. Appl.,2(5), 436–451.
Walter, G.G. (1992). A sampling theorem for wavelet subspaces,IEEE Trans. Inform. Theory, part 2,116(2) 881–884.
Xia, X.G. and Zhang, Z.Z. (1993). On sampling theorem, wavelets, and wavelet transforms,IEEE Trans. Signal Process.,41(12), 3524–3535.
Yao, K. (1967). Applications of reproducing kernel Hilbert spaces-bandlimited signal models,Information and Control,11, 429–444.
Author information
Authors and Affiliations
Additional information
Communicated by John J. Benedetto
Rights and permissions
About this article
Cite this article
Aldroubi, A., Gröchenig, K. Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces. The Journal of Fourier Analysis and Applications 6, 93–103 (2000). https://doi.org/10.1007/BF02510120
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02510120