A $(p,q)$ version of Bourgain's theorem

Let 1 < p, q < ∞ satisfy 1 p + 1 q = 1. We construct an orthonormal basis {b n } for L 2 (R) such that Δp(b n ) and Δ q (bn) are both uniformly bounded in n. Here Δ λ (f) ≡ inf α∈R (|x-a|λ|f(x)| 2 dx) 1 2 . This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.

[1]  H. Feichtinger,et al.  Gabor Frames and Time-Frequency Analysis of Distributions* , 1997 .

[2]  Thomas Strohmer.  Four short stories about Toeplitz matrix calculations , 2000 .

[3]  F. Low.  Complete sets of wave packets , 1985 .

[4]  W. Rudin.  New constructions of functions holomorphic in the unit ball of C [italic superscript n] , 1986 .

[5]  H. Feichtinger,et al.  Banach spaces related to integrable group representations and their atomic decompositions, I , 1989 .

[6]  Deguang Han,et al.  The Balian–Low theorem for symplectic lattices in higher dimensions , 2002 .

[7]  O. Christensen.  Frames, Riesz bases, and discrete Gabor/wavelet expansions , 2001 .

[8]  Chung-I Tan,et al.  A Passion for Physics: Essays in Honor of Geoffrey Chew , 1986 .

[9]  Deguang Han,et al.  Balian–Low phenomenon for subspace Gabor frames , 2004 .

[10]  R. Balian.  Un principe d'incertitude fort en théorie du signal ou en mécanique quantique , 1981 .

[11]  Christopher Heil,et al.  Linear independence of time-frequency translates , 1996 .

[12]  M. Cowling,et al.  Bandwidth Versus Time Concentration: The Heisenberg–Pauli–Weyl Inequality , 1984 .

[13]  K. Gröchenig.  An uncertainty principle related to the Poisson summation formula , 1996 .

[14]  G. Folland,et al.  The uncertainty principle: A mathematical survey , 1997 .

[15]  John J. Benedetto,et al.  The Balian–Low theorem for the symplectic form on R2d , 2003 .

[16]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[17]  Y. Meyer,et al.  Ondelettes et bases hilbertiennes. , 1986 .

[18]  G. Folland.  Harmonic analysis in phase space , 1989 .

[19]  Jean Bourgain,et al.  A remark on the uncertainty principle for Hilbertian basis , 1988 .

[20]  G. Folland.  Harmonic Analysis in Phase Space. (AM-122), Volume 122 , 1989 .

[21]  Karlheinz Gröchenig,et al.  Foundations of Time-Frequency Analysis , 2000, Applied and numerical harmonic analysis.

[22]  V. Havin.  The Uncertainty Principle in Harmonic Analysis , 1994 .

[23]  John J. Benedetto,et al.  The Balian-Low theorem and regularity of Gabor systems , 2003 .

[24]  V. Guillemin,et al.  Review: Gerald Folland, Harmonic analysis in phase space , 1990 .

[25]  Stéphane Jaffard.  Propriétés des matrices « bien localisées » près de leur diagonale et quelques applications , 1990 .

[26]  Ingrid Daubechies,et al.  Two theorems on lattice expansions , 1993, IEEE Trans. Inf. Theory.

[27]  K. Hannabuss,et al.  HARMONIC ANALYSIS IN PHASE SPACE: (Annals of Mathematics Studies 122) , 1990 .