## Reconstruction algorithms in irregular sampling

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- by Karlheinz Gröchenig PDF
- Math. Comp.
**59**(1992), 181-194 Request permission

## Abstract:

A constructive solution of the irregular sampling problem for band-limited functions is given. We show how a band-limited function can be completely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions.## References

- Arne Beurling,
*Local harmonic analysis with some applications to differential operators*, Some Recent Advances in the Basic Sciences, Vol. 1 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1962–1964) Belfer Graduate School of Science, Yeshiva Univ., New York, 1966, pp. 109–125. MR**0427956** - Arne Beurling and Paul Malliavin,
*On the closure of characters and the zeros of entire functions*, Acta Math.**118**(1967), 79–93. MR**209758**, DOI 10.1007/BF02392477 - Paul L. Butzer and Guido Hinsen,
*Two-dimensional nonuniform sampling expansions—an iterative approach. II. Reconstruction formulae and applications*, Appl. Anal.**32**(1989), no. 1, 69–85. MR**1017524**, DOI 10.1080/00036818908839839 - P. L. Butzer, W. Splettstösser, and R. L. Stens,
*The sampling theorem and linear prediction in signal analysis*, Jahresber. Deutsch. Math.-Verein.**90**(1988), no. 1, 70. MR**928745** - W. J. Coles,
*A general Wirtinger-type inequality*, Duke Math. J.**27**(1960), 133–138. MR**110770**, DOI 10.1215/S0012-7094-60-02713-7 - R. J. Duffin and A. C. Schaeffer,
*A class of nonharmonic Fourier series*, Trans. Amer. Math. Soc.**72**(1952), 341–366. MR**47179**, DOI 10.1090/S0002-9947-1952-0047179-6 - Ky Fan, Olga Taussky, and John Todd,
*Discrete analogs of inequalities of Wirtinger*, Monatsh. Math.**59**(1955), 73–90. MR**70676**, DOI 10.1007/BF01302991 - Hans G. Feichtinger,
*Discretization of convolution and reconstruction of band-limited functions from irregular sampling*, Progress in approximation theory, Academic Press, Boston, MA, 1991, pp. 333–345. MR**1114782** - 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 - Hans G. Feichtinger and Karlheinz Gröchenig,
*Irregular sampling theorems and series expansions of band-limited functions*, J. Math. Anal. Appl.**167**(1992), no. 2, 530–556. MR**1168605**, DOI 10.1016/0022-247X(92)90223-Z
—, - Hans G. Feichtinger and Karlheinz Gröchenig,
*Error analysis in regular and irregular sampling theory*, Appl. Anal.**50**(1993), no. 3-4, 167–189. MR**1278324**, DOI 10.1080/00036819308840192
H. G. Feichtinger, K. Gröchenig, and M. Hermann, - Karlheinz Gröchenig,
*A new approach to irregular sampling of band-limited functions*, Recent advances in Fourier analysis and its applications (Il Ciocco, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 315, Kluwer Acad. Publ., Dordrecht, 1990, pp. 251–260. MR**1081352**
G. Hardy, J. E. Littlewood, and G. Pólya, - H. J. Landau,
*Necessary density conditions for sampling and interpolation of certain entire functions*, Acta Math.**117**(1967), 37–52. MR**222554**, DOI 10.1007/BF02395039
—, - Norman Levinson,
*Gap and Density Theorems*, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940. MR**0003208**, DOI 10.1090/coll/026
F. A. Marvasti, - Michael D. Rawn,
*A stable nonuniform sampling expansion involving derivatives*, IEEE Trans. Inform. Theory**35**(1989), no. 6, 1223–1227. MR**1036626**, DOI 10.1109/18.45278
K. D. Sauer and J. P. Allebach, - Robert M. Young,
*An introduction to nonharmonic Fourier series*, 1st ed., Academic Press, Inc., San Diego, CA, 2001. MR**1836633**

*Multidimensional irregular sampling of band-limited functions in*${L^p}$-

*spaces, in Multivariate Approximation Theory*IV (C. K. Chui, W. Schempp and K. Zeller, eds.), ISNM

**90**, Birkhäuser, Basel, 1989, pp. 135-142.

*Iterative methods in irregular sampling theory*, Numerical Results, vol. 7, Aachener Symposium für Signaltheorie, ASST 1990, Aachen, Informatik Fachber. 253, Springer, 1990, pp. 160-166.

*Inequalities*, 2nd ed., Cambridge Univ. Press, 1952. A. J. Jerri,

*The Shannon sampling theorem—its various extensions and applications. A tutorial review*, Proc. IEEE

**65**(1977), 1565-1596.

*Sampling data transmission and the Nyquist rate*, Proc. IEEE

**55**(1967), 1701-1706.

*A unified approach to zero-crossing and nonuniform sampling of single and multidimensional systems*, Nonuniform (P. O. Box 1505, Oak Park, IL 60304), 1987. F. Marvasti and M. Analoui,

*Recovery of signals from nonuniform samples using iterative methods*, Proc. Internat. Sympos. Circuits Systems, Portland, OR, May 1989. F. A. Marvasti,

*An iterative method to compensate for the interpolation distortion*, IEEE Trans. Acoust. Speech Signal Process.

**37**(1989), 1619-1621. A. Papoulis,

*Signal analysis*, McGraw-Hill, New York, 1977.

*Iterative reconstruction of band-limited images from nonuniformly spaced samples*, IEEE Trans. Circuits and Systems

**34**(1987), 1497-1506. R. G. Wiley,

*Recovery of band-limited signals from unequally spaced samples*, IEEE Trans. Comm.

**26**(1978), 135-137. S. Yeh and H. Stark,

*Iterative and one-step reconstruction from nonuniform samples by convex projections*, J. Opt. Soc. Amer. A

**7**(1990), 491-499. D. C. Youla and H. Webb,

*Image restoration by the method of convex projections*:

*Part*I, IEEE Trans. Med. Imag.

**1**(1982), 81-94.

## Additional Information

- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp.
**59**(1992), 181-194 - MSC: Primary 41A25; Secondary 41A80, 42A15, 65D99, 94A12
- DOI: https://doi.org/10.1090/S0025-5718-1992-1134729-0
- MathSciNet review: 1134729