Time-frequency concentration of generating systems

Jaming, Philippe; Powell, Alexander

doi:10.1090/S0002-9939-2011-10768-8

Time-frequency concentration of generating systems
HTML articles powered by AMS MathViewer

by Philippe Jaming and Alexander M. Powell PDF

Proc. Amer. Math. Soc. 139 (2011), 3279-3290 Request permission

Abstract:

Uncertainty principles for generating systems $\{e_n\}_{n=1}^{\infty } \subset L^2(\mathbb {R})$ are proven and quantify the interplay between $\ell ^r(\mathbb {N})$ coefficient stability properties and time-frequency localization with respect to $|t|^p$ power weight dispersions. As a sample result, it is proven that if the unit-norm system $\{e_n\}_{n=1}^{\infty }$ is a Schauder basis or frame for $L^2(\mathbb {R})$, then the two dispersion sequences $\Delta (e_n)$, $\Delta (\widehat {e_n})$ and the one mean sequence $\mu (e_n)$ cannot all be bounded. On the other hand, it is constructively proven that there exists a unit-norm exact system $\{f_n\}_{n=1}^{\infty }$ in $L^2(\mathbb {R})$ for which all four of the sequences $\Delta (f_n)$, $\Delta (\widehat {f_n})$, $\mu (f_n)$, $\mu (\widehat {f_n})$ are bounded.

References

Gerard Ascensi, Yurii Lyubarskii, and Kristian Seip, Phase space distribution of Gabor expansions, Appl. Comput. Harmon. Anal. 26 (2009), no. 2, 277–282. MR 2490219, DOI 10.1016/j.acha.2008.07.005
John J. Benedetto, Christopher Heil, and David F. Walnut, Differentiation and the Balian-Low theorem, J. Fourier Anal. Appl. 1 (1995), no. 4, 355–402. MR 1350699, DOI 10.1007/s00041-001-4016-5
J. Bourgain, A remark on the uncertainty principle for Hilbertian basis, J. Funct. Anal. 79 (1988), no. 1, 136–143. MR 950087, DOI 10.1016/0022-1236(88)90033-X
Peter G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000), no. 2, 129–201. MR 1757401, DOI 10.11650/twjm/1500407227
Pete Casazza, Ole Christensen, and Diana T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl. 307 (2005), no. 2, 710–723. MR 2142455, DOI 10.1016/j.jmaa.2005.02.015
Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1946982, DOI 10.1007/978-0-8176-8224-8
Monika Dörfler, Hans G. Feichtinger, and Karlheinz Gröchenig, Compactness criteria in function spaces, Colloq. Math. 94 (2002), no. 1, 37–50. MR 1930200, DOI 10.4064/cm94-1-3
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. MR 1448337, DOI 10.1007/BF02649110
C. Heil and A. M. Powell, Regularity for complete and minimal Gabor systems on a lattice, Illinois J. Math., 53 (2009), 1077–1094.
H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Expo. Math. 28 (2010), 385–394.
V. I. Gurariĭ and N. I. Gurariĭ, Bases in uniformly convex and uniformly smooth Banach spaces, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 210–215 (Russian). MR 0283549
Philippe Jaming and Alexander M. Powell, Uncertainty principles for orthonormal sequences, J. Funct. Anal. 243 (2007), no. 2, 611–630. MR 2289698, DOI 10.1016/j.jfa.2006.09.001
A. N. Kolmogorov, Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel, Nachr. Ges. Wiss. Göttingen 9 (1931), 60–63.
E. Malinnikova, Orthonormal sequences in $L^2(R^d)$ and time frequency localization, Jour. Fourier. Anal. Appl. 16 (2010), 983–1006.
S. Nitzan and J.-F. Olsen, From exact sequences to Riesz bases in the Balian-Low theorem, accepted to appear in Jour. Fourier Anal. Appl., DOI 10.100/s00041-010-9150-5.
Robert L. Pego, Compactness in $L^2$ and the Fourier transform, Proc. Amer. Math. Soc. 95 (1985), no. 2, 252–254. MR 801333, DOI 10.1090/S0002-9939-1985-0801333-9
Alexander M. Powell, Time-frequency mean and variance sequences of orthonormal bases, J. Fourier Anal. Appl. 11 (2005), no. 4, 375–387. MR 2169472, DOI 10.1007/s00041-005-3082-5
M. Riesz, Sur les ensembles compacts de fonctions sommables, Acta Szeged Sect. Math. 6 (1933), 136–142.
William H. Ruckle, The extent of the sequence space associated with a basis, Canadian J. Math. 24 (1972), 636–641. MR 300057, DOI 10.4153/CJM-1972-059-8
H. S. Shapiro, Uncertainty principles for bases in $L^2(\mathbb {R})$, unpublished manuscript (1991).
J. D. Tamarkin, On the compactness of the space $L_p$, Bull. Amer. Math. Soc. 38 (1932), no. 2, 79–84. MR 1562331, DOI 10.1090/S0002-9904-1932-05332-0
Robert M. Young, An introduction to nonharmonic Fourier series, 1st ed., Academic Press, Inc., San Diego, CA, 2001. MR 1836633
Robert M. Young, On complete biorthogonal systems, Proc. Amer. Math. Soc. 83 (1981), no. 3, 537–540. MR 627686, DOI 10.1090/S0002-9939-1981-0627686-9
Kôsaku Yosida, Functional analysis, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York, Inc., New York, 1968. MR 0239384, DOI 10.1007/978-3-662-11791-0