A Note on Rearrangements, Spectral Concentration, and the Zero-Order Prolate Spheroidal Wavefunction

IEEE Transactions on Information Theory

Volumn 39, Issue 1, 1993, Pages 257-260

  Donoho, David L ^a   Stark, Philip B ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

prolate spheroidal wavefunctions; smoothing; sparsity constraints; Symmetric decreasing rearrangement; uncertainty principle

Indexed keywords

FOURIER TRANSFORMS; OPTIMIZATION; SPECTRUM ANALYSIS;

EXPLORATION SEISMOLOGY; PROLATE SPHEROIDAL WAVEFUNCTION; SMOOTHING; SPARSITY CONSTRAINTS; SYMMETRIC DECREASING REARRANGEMENT; UNCERTAINTY PRINCIPLE;

SEISMOLOGY;

EID: 0027222378     PISSN: 00189448     EISSN: 15579654     Source Type: Journal    
DOI: 10.1109/18.179370     Document Type: Article

References (22)

1. M. Abramowitz, and I. A. Stegun, Eds., Handbook of Mathematical Functions. New York: Dover, 1965.
2. D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math., vol. 49, pp. 906–931, 1989.
3. —, “Rearrangements and smoothing,” Tech. Rep. 148, Dept. of Statist., Univ. of California, Berkeley, 1988.
4. A. M. Garsia, “Combinatorial inequalities and smoothness of functions,” Bull. Am. Math. Soc., vol. 82, pp. 157–170, 1976.
5. G. H. Hardy, and J. E. Littlewood, “Notes on the theory of series (XIII): Some new properties of Fourier constants,” J. Lon. Math. Soc., vol. 6, pp. 3–9, 1931.
6. —, “A new proof of a theorem on rearrangements,” J. Lon. Math. Soc., vol. 23, pp. 163–168, 1948.
7. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities. New York: Cambridge Univ. Press, 1978.
8. W. B. Jurkat, and G. Sampson, “On maximal rearrangement inequalities for the Fourier transform,” Trans. Am. Math. Soc., vol. 282, pp. 625–643, 1984.
9. —, “On rearrangement and weight inequalities for the Fourier transform,” Ind. U. Math. J., vol. 33, pp. 257–270, 1984.
10. B. Kawohl, Rearrangement and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, vol. 1150. New York: Springer-Verlag, 1985.
11. F. R. Keogh, “On some inequalities of Littlewood and a problem on rearrangements,” J. Lon. Math. Soc., vol. 36, pp. 362–370, 1961.
12. H. J. Landau, and H. O. Pollak, “Prolate spheroidal wavefunctions, Fourier analysis, and uncertainty—II,” Bell Syst. Tech. J., vol. 40, pp. 64–84, 1961.
13. E.H. Lieb, “Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,” Ann. Math., vol. 118, pp. 349–374, 1983.
14. J. E. Littlewood, “On a theorem of Paley,” J. Lon. Math. Soc., vol. 29, pp. 387–395, 1954.
15. —, “On the inequalities between functions f and f*,” J. Lon. Math. Soc., vol. 35, pp. 352–365, 1960.
16. H.L. Montgomery, “A note on rearrangements of Fourier coefficients,” Ann. Inst. Fourier, Grenoble, vol. 26, pp. 2, 29–34, 1976.
17. B. Muchenhoupt, “Weighted norm inequalities for the Fourier transform,” Trans. Am. Math. Soc., vol. 276, pp. 729–742, 1983.
18. D. W. Oldenburg, T. Scheuer, and S. Levy, “Recovery of the acoustic impedance from reflection seismograms,” Geophys., vol. 48, pp. 1318–1337, 1983.
19. F. Riesz, “Sur une inegalite integrate,” J. Math. Soc., vol. 5, pp. 162–168, 1930.
20. F. Santosa, and W. W. Symes, “Linear inversion of band-limited reflection seismograms,” SIAM J. Sci. Statist. Comput., vol. 7, pp. 1307–1330, 1986.
21. H. S. Shapiro, “Majorant problems for Fourier coefficients,” Quart. J. Math. Oxford, vol. 26, no. 2, pp. 9–19, 1975.
22. D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J., vol. 40, pp. 43–63, 1961.