## Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The $L^p$-theory

HTML articles powered by AMS MathViewer

- by Akram Aldroubi and Hans Feichtinger PDF
- Proc. Amer. Math. Soc.
**126**(1998), 2677-2686 Request permission

## Abstract:

We prove that the exact reconstruction of a function $s$ from its samples $s (x_i)$ on any “sufficiently dense" sampling set $\{x_i\}_{i\in \Lambda }$ can be obtained, as long as $s$ is known to belong to a large class of spline-like spaces in $L^p (\mathcal {R}^n)$. Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling.## References

- Akram Aldroubi,
*Oblique projections in atomic spaces*, Proc. Amer. Math. Soc.**124**(1996), no. 7, 2051–2060. MR**1317028**, DOI 10.1090/S0002-9939-96-03255-8 - Akram Aldroubi and Michael Unser,
*Families of wavelet transforms in connection with Shannon’s sampling theory and the Gabor transform*, Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 509–528. MR**1161261** - Akram Aldroubi and Michael Unser,
*Families of multiresolution and wavelet spaces with optimal properties*, Numer. Funct. Anal. Optim.**14**(1993), no. 5-6, 417–446. MR**1248121**, DOI 10.1080/01630569308816532 - Akram Aldroubi and Michael Unser,
*Sampling procedures in function spaces and asymptotic equivalence with Shannon’s sampling theory*, Numer. Funct. Anal. Optim.**15**(1994), no. 1-2, 1–21. MR**1261594**, DOI 10.1080/01630569408816545 - A. Aldroubi, M. Unser, and M. Eden. Cardinal spline filters: Stability and convergence to the ideal sinc interpolator.
*Signal Processing*, 28:127–138, 1992. - John J. Benedetto,
*Irregular sampling and frames*, Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 445–507. MR**1161260** - J. J. Corliss,
*Upper limits to the real roots of a real algebraic equation*, Amer. Math. Monthly**46**(1939), 334–338. MR**4**, DOI 10.1080/00029890.1939.11998880 - H. G. Feichtinger,
*Banach convolution algebras of Wiener type*, Functions, series, operators, Vol. I, II (Budapest, 1980) Colloq. Math. Soc. János Bolyai, vol. 35, North-Holland, Amsterdam, 1983, pp. 509–524. MR**751019** - Hans G. Feichtinger,
*Generalized amalgams, with applications to Fourier transform*, Canad. J. Math.**42**(1990), no. 3, 395–409. MR**1062738**, DOI 10.4153/CJM-1990-022-6 - Hans G. Feichtinger,
*New results on regular and irregular sampling based on Wiener amalgams*, Function spaces (Edwardsville, IL, 1990) Lecture Notes in Pure and Appl. Math., vol. 136, Dekker, New York, 1992, pp. 107–121. MR**1152342** - Hans G. Feichtinger,
*New results on regular and irregular sampling based on Wiener amalgams*, Function spaces (Edwardsville, IL, 1990) Lecture Notes in Pure and Appl. Math., vol. 136, Dekker, New York, 1992, pp. 107–121. MR**1152342** - H. G. Feichtinger and K. Gröchenig,
*Nonorthogonal wavelet and Gabor expansions, and group representations*, Wavelets and their applications, Jones and Bartlett, Boston, MA, 1992, pp. 353–375. MR**1187349** - Hans G. Feichtinger and Karlheinz Gröchenig,
*Iterative reconstruction of multivariate band-limited functions from irregular sampling values*, SIAM J. Math. Anal.**23**(1992), no. 1, 244–261. MR**1145171**, DOI 10.1137/0523013 - J. R. Higgins,
*Five short stories about the cardinal series*, Bull. Amer. Math. Soc. (N.S.)**12**(1985), no. 1, 45–89. MR**766960**, DOI 10.1090/S0273-0979-1985-15293-0 - A.J. Jerri. The Shannon sampling theorem—its various extensions and applications: A tutorial review.
*Proc. IEEE*, 65:1565–1596, 1977. - M. Ĭ. Kadec′,
*The exact value of the Paley-Wiener constant*, Dokl. Akad. Nauk SSSR**155**(1964), 1253–1254 (Russian). MR**0162088** - Joram Lindenstrauss and Lior Tzafriri,
*Classical Banach spaces*, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR**0415253** - Youming Liu,
*Irregular sampling for spline wavelet subspaces*, IEEE Trans. Inform. Theory**42**(1996), no. 2, 623–627. MR**1381369**, DOI 10.1109/18.485731 - R.E.A.C Paley and N. Wiener. Fourier transform in the complex domain.
*Amer. Math. Soc. Colloquium publications*. Amer. Math. Soc., 1934. - Morgan Ward,
*Ring homomorphisms which are also lattice homomorphisms*, Amer. J. Math.**61**(1939), 783–787. MR**10**, DOI 10.2307/2371336 - M. Unser and A. Aldroubi. A general sampling theory for non-ideal acquisition devices.
*IEEE Trans. on Signal Processing*, 42(11):2915–2925, 1994. - M. Unser, A. Aldroubi, and M. Eden. Polynomial spline signal approximations: filter design and asymptotic equivalence with Shannon’s sampling theorem.
*IEEE Trans. Image Process.*, 38:95–103, 1991. - J.M. Whittaker.
*Interpolation Function Theory*. Cambridge University Press, London, 1935. - K. Yao. Application of Reproducing Kernel Hilbert Spaces—bandlimited signal models.
*Information and Control*, 11:429–444, 1967. - Ahmed I. Zayed,
*On Kramer’s sampling theorem associated with general Sturm-Liouville problems and Lagrange interpolation*, SIAM J. Appl. Math.**51**(1991), no. 2, 575–604. MR**1095036**, DOI 10.1137/0151030

## Additional Information

**Akram Aldroubi**- Affiliation: National Institutes of Health, Biomedical Engineering and Instrumentation Program, Bethesda, Maryland 20892
- Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- Email: aldroubi@helix.nih.gov, aldroubi@math.vanderbilt.edu
**Hans Feichtinger**- Affiliation: University of Vienna, Department of Mathematics, Strudlhofg. 4, A-1090 Wien, Austria
- MR Author ID: 65680
- ORCID: 0000-0002-9927-0742
- Email: fei@tyche.mat.univie.ac.at
- Received by editor(s): January 28, 1997
- Additional Notes: This research was partially supported through the FWF-project S-7001-MAT of the Austrian Science Foundation.
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**126**(1998), 2677-2686 - MSC (1991): Primary 42C15, 46A35, 46E15, 46N99, 47B37
- DOI: https://doi.org/10.1090/S0002-9939-98-04319-6
- MathSciNet review: 1451788

Dedicated: Dedicated to the memory of Richard J. Duffin