Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces
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 B1/2. 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 B1/2.
Math Subject Classifications42C15 42A65
Keywords and Phrasesirregular sampling spline shift invariant spaces frames Wiener amalgam spaces reproducing Kernel Hilbert spaces
Unable to display preview. Download preview PDF.
- 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.Google Scholar
- 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.Google Scholar
- Beurling, A. (1989). In Carleson, L., Ed.,A. Beurling. Collected Works. Vol. 2, Birkhäuser, Boston, 341–365.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- Mallat, S. (1992). Personal communication, conf. oberwolfach, wavelets and applications.Google Scholar
- Schumaker, L. (1980).Spline Functions: Basic Theory. Wiley-Interscience, Boston.Google Scholar