Flexible Gabor-wavelet atomic decompositions for L2-Sobolev spaces

In this paper we present a general construction of frames, which allows one to ensure that certain families of functions (atoms) obtained by a suitable combination of translation, modulation, and dilation will form Banach frames for the family of L2-Sobolev spaces on ℝ of any order. In this construction a parameter α∈[0,1) governs the dependence of the dilation factor on the frequency parameter. The well-known Gabor and wavelet frames (also valid for the same scale of Hilbert spaces) using suitable Schwartz functions as building blocks arise as special cases (α=0) and a limiting case (α→1), respectively. In contrast to those limiting cases, it is no longer possible to use group-theoretical arguments. Nevertheless, we will show how to constructively ensure that for Schwartz analyzing atoms and any sufficiently dense but discrete and well-structured family of parameters one can guarantee the frame property. As a consequence of this novel constructive technique, one can generate quasicoherent dual frames by an iterative algorithm. As will be shown in a subsequent paper, the new frames introduced here generate Banach frames for corresponding families of α-modulation spaces.

[1]  H. Feichtinger.  On a new Segal algebra , 1981 .

[2]  O. Christensen,et al.  Approximation of the Inverse Frame Operator and Applications to Gabor Frames , 2000 .

[3]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[4]  K. Gröchenig.  Describing functions: Atomic decompositions versus frames , 1991 .

[5]  B. Torrésani.  Wavelets associated with representations of the affine Weyl–Heisenberg group , 1991 .

[6]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[7]  Gabriele Steidl,et al.  Weighted Coorbit Spaces and Banach Frames on Homogeneous Spaces , 2004 .

[8]  H. Triebel.  Theory of Function Spaces III , 2008 .

[9]  T. Strohmer,et al.  Efficient numerical methods in non-uniform sampling theory , 1995 .

[10]  A. Grossmann,et al.  Transforms associated to square integrable group representations. I. General results , 1985 .

[11]  Hans G. Feichtinger,et al.  Wiener Amalgams over Euclidean Spaces and Some of Their Applications , 2020 .

[12]  Massimo Fornasier,et al.  Banach frames for α-modulation spaces , 2007 .

[13]  B. Torrésani.  Time-frequency representations : wavelet packets and optimal decomposition , 1992 .

[14]  Massimo Fornasier,et al.  Intrinsic Localization of Frames , 2005 .

[15]  H. Feichtinger,et al.  Atomic Systems for Subspaces , 2001 .

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

[17]  H. Feichtinger,et al.  Irregular sampling theorems and series expansions of band-limited functions , 1992 .

[18]  H. Feichtinger,et al.  Iterative reconstruction of multivariate band-limited functions from irregular sampling values , 1992 .

[19]  H. Triebel.  Theory Of Function Spaces , 1983 .

[20]  K. Gröchenig.  Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator , 2004 .

[21]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

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

[23]  H. Feichtinger.  Atomic characterizations of modulation spaces through Gabor-type representations , 1989 .

[24]  H. Feichtinger,et al.  Banach Spaces of Distributions Defined by Decomposition Methods, I , 1985 .

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

[26]  M. Holschneider,et al.  An Interpolation Family between Gabor and Wavelet Transformations , 2003 .

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

[28]  Charles Fefferman,et al.  Wave packets and fourier integral operators , 1978 .

[29]  H. Feichtinger.  Generalized Amalgams, With Applications to Fourier Transform , 1990, Canadian Journal of Mathematics.

[30]  Joseph D. Lakey,et al.  Extensions of the Heisenberg Group by Dilations and Frames , 1995 .

[31]  Syed Twareque Ali,et al.  Continuous Frames in Hilbert Space , 1993 .

[32]  H. Feichtinger,et al.  A unified approach to atomic decompositions via integrable group representations , 1988 .

[33]  S. Albeverio,et al.  Advances in Partial differential equations , 2001 .

[34]  Bruno Torrésani,et al.  Hybrid representations for audiophonic signal encoding , 2002, Signal Process..

[35]  K. Chung.  MATHEMATICS AND APPLICATIONS , 2004 .

[36]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[37]  M. Fornasier.  Quasi-orthogonal decompositions of structured frames , 2004 .

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

[39]  C. Heil.  An Introduction to Weighted Wiener Amalgams , 2003 .