## Illinois Journal of Mathematics

### Relevant sampling of band-limited functions

#### Abstract

We study the random sampling of band-limited functions of several variables. If a band-limited function with bandwidth has its essential support on a cube of volume $R^{d}$, then $\mathcal{O}(R^{d}\log R^{d})$ random samples suffice to approximate the function up to a given error with high probability.

#### Article information

Source
Illinois J. Math., Volume 57, Number 1 (2013), 43-58.

Dates
First available in Project Euclid: 23 June 2014

https://projecteuclid.org/euclid.ijm/1403534485

Digital Object Identifier
doi:10.1215/ijm/1403534485

Mathematical Reviews number (MathSciNet)
MR3224560

Zentralblatt MATH identifier
1372.94378

#### Citation

Bass, Richard F.; Gröchenig, Karlheinz. Relevant sampling of band-limited functions. Illinois J. Math. 57 (2013), no. 1, 43--58. doi:10.1215/ijm/1403534485. https://projecteuclid.org/euclid.ijm/1403534485

#### References

• R. Ahlswede and A. Winter, Strong converse for identification via quantum channels, IEEE Trans. Inform. Theory 48 (2002), no. 3, 569–579.
• R. F. Bass and K. Gröchenig, Random sampling of band-limited functions, Israel J. Math. 177 (2010), 1–28.
• A. 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.
• S. Foucard and H. Rauhut, A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013.
• K. Gröchenig, Reconstruction algorithms in irregular sampling, Math. Comp. 59 (1992), no. 199, 181–194.
• H. Landau, On the density of phase space expansions, IEEE Trans. Inform. Theory 39 (1993), 1152–1156.
• H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37–52.
• H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. II, Bell System Tech. J. 40 (1961), 65–84.
• H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. III. The dimension of the space of essentially time- and band-limited signals, Bell System Tech. J. 41 (1962), 1295–1336.
• B. Matei and Y. Meyer, Simple quasicrystals are sets of stable sampling, Complex Var. Elliptic Equ. 55 (2010), no. 8–10, 947–964.
• S. Mendelson and A. Pajor, On singular values of matrices with independent rows, Bernoulli 12 (2006), no. 5, 761–773.
• A. Olevskiĭ and A. Ulanovskii, Universal sampling and interpolation of band-limited signals, Geom. Funct. Anal. 18 (2008), no. 3, 1029–1052.
• R. I. Oliveira, Sums of random Hermitian matrices and an inequality by Rudelson, Electron. Commun. Probab. 15 (2010), 203–212.
• J. Ortega-Cerdà and K. Seip, Fourier frames, Ann. of Math. (2) 155 (2002), no. 3, 789–806.
• M. Rudelson, Random vectors in the isotropic position, J. Funct. Anal. 164 (1999), no. 1, 60–72.
• M. Rudelson and R. Vershynin, Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62 (2009), no. 12, 1707–1739.
• D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J. 43 (1964), 3009–3057.
• D. Slepian, On bandwidth, Proc. IEEE 64 (1976), no. 3, 292–300.
• D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. I, Bell System Tech. J. 40 (1961), 43–63.
• S. Smale and D.-X. Zhou, Shannon sampling and function reconstruction from point values, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 3, 279–305 (electronic).
• H. Triebel, Theory of function spaces, Birkhäuser, Basel, 1983.
• J. Tropp, User-friendly tail bounds for sums of random matrices, Found. Comput. Math. 12 (2012), 389–434.
• R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, Compressed sensing, theory and applications (Y. Eldar, and G. Kutyniok, eds.), Cambridge Univ. Press, Cambridge, 2012, Chapter 5, pp. 210–268.
• H. Widom, Asymptotic behavior of the eigenvalues of certain integral equations. II, Arch. Rational Mech. Anal. 17 (1964), 215–229.