The geometry of sets of parameters of wave packet frames

We study wave packet systems WP(ψ,M); that is, countable collections of dilations, translations, and modulations of a single function ψ ∈ L 2 (R).The parameters of these unitary actions form a discrete subset M⊂ R + × R × R.We introduce analogues of the notion of Beurling density, adapted to the geometry of discrete subsets of R + × R × R, and notions of lower and upper dimensions associated with these densities.Our goal is to describe completeness properties of wave packet systems via geometric properties of the sets of their parameters.In particular, we show necessary conditions for WP(ψ,M) to be a Bessel system, and we construct multiple examples of non-standard wave packet frames with prescribed dimensions.

[1]  G. Kutyniok,et al.  Density of weighted wavelet frames , 2003 .

[2]  Lp bounds for a maximal dyadic sum operator , 2002, math/0212164.

[3]  Hans G. Feichtinger,et al.  Flexible Gabor-wavelet atomic decompositions for L2-Sobolev spaces , 2006 .

[4]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[5]  M. Lacey,et al.  A proof of boundedness of the Carleson operator , 2000 .

[6]  O. Christensen,et al.  Density of Gabor Frames , 1999 .

[7]  Christoph Thiele,et al.  On Calderon s conjecture , 1999 .

[8]  Michael Lacey,et al.  L P Estimates on the Bilinear Hilbert Transform for 2 < P < 1 , 1997 .

[9]  Stephen Taylor,et al.  Fractional dimension of sets in discrete spaces , 1989 .

[10]  A. Olevskiǐ,et al.  Almost Integer Translates. Do Nice Generators Exist? , 2004 .

[11]  On Calder\'on's conjecture , 1999, math/9903203.

[12]  Demetrio Labate,et al.  A unified characterization of reproducing systems generated by a finite family, II , 2002 .

[13]  A. Atzmon,et al.  Completeness of Integer Translates in Function Spaces on R , 1996 .

[14]  Demetrio Labate,et al.  A unified characterization of reproducing systems generated by a finite family , 2002 .

[15]  Demetrio Labate,et al.  An Approach to the Study of Wave Packet Systems , 2003 .

[16]  O. Christensen,et al.  Irregular Wavelet Frames and Gabor Frames , 2001 .

[17]  Demetrio Labate,et al.  Oversampling, quasi-affine frames, and wave packets , 2004 .

[18]  Christoph Thiele,et al.  $L^p$ estimates on the bilinear Hilbert transform for $2 < p < \infty$ , 1997 .