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Weyl-Heisenberg frames and Riesz bases in $$L_2(\mathbb{R}^d)$$. (English) Zbl 0892.42017
From the introduction: “The present paper is the second in a series of three, all devoted to the study of shift-invariant frames and shift-invariant stable (= Riesz) bases for $$H:= L_2(\mathbb{R}^d)$$, $$d\geq 1$$, or a subspace of it. In the first paper [RS1] [Can. J. Math. 47, No. 5, 1051-1094 (1995; Zbl 0838.42016)], we studied such bases under the mere assumption that the basis set can be written as a collection of shifts (namely, integer translates) of a set of generators $$\Phi$$. The present paper analyzes Weyl-Heisenberg (WH, known also as Gaborian) frames and stable bases. Aside from specializing the general methods and results of [RS1] to his important case, we exploit here the special structure of the WH set, and in particular the duality between the shift operator and the modulation operator, the latter being absent in the context of general shift-invariant sets. In the third paper [RS3] [J. Funct. Anal. 148, No. 2, 408-447 (1997; Zbl 0891.42018)], we present applications of the results of [RS1] to wavelet (or affine) frames. The flavour of the results there is quite different; wavelet sets are not shift-invariant, and the main effort of [RS3] is to show that, nevertheless, the basic analysis of [RS1] does apply to that case as well”.

##### MSC:
 42C15 General harmonic expansions, frames
Full Text:
##### References:
 [1] J. J. Benedetto and D. F. Walnut, Gabor frames for $$L^ 2$$ and related spaces , Wavelets: Mathematics and Applications eds. J. Benedetto and M. Frazier, Stud. Adv. Math., CRC, Boca Raton, FLa, 1994, pp. 97-162. · Zbl 0887.42025 [2] C. K. Chui, An Introduction to Wavelets , Wavelet Analysis and its Applications, vol. 1, Academic Press Inc., Boston, MA, 1992. · Zbl 0925.42016 [3] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis , IEEE Trans. Inform. Theory 36 (1990), no. 5, 961-1005. · Zbl 0738.94004 [4] I. Daubechies, Ten lectures on wavelets , CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. · Zbl 0776.42018 [5] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions , J. Math. Phys. 27 (1986), no. 5, 1271-1283. · Zbl 0608.46014 [6] I. Daubechies, H. J. Landau, and Z. Landau, Gabor time-frequency lattices and the Wexler-Raz identity , J. Fourier Anal. Appl. 1 (1995), no. 4, 437-478. · Zbl 0888.47018 [7] C. de Boor, R. DeVore, and A. Ron, The structure of finitely generated shift-invariant spaces in $$L_ 2(\mathbf R^ d)$$ , J. Funct. Anal. 119 (1994), no. 1, 37-78, FTP site anonymous@stolp.cs.wisc.edu, file name several.ps. · Zbl 0806.46030 [8] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series , Trans. Amer. Math. Soc. 72 (1952), 341-366. JSTOR: · Zbl 0049.32401 [9] H. G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I , J. Funct. Anal. 86 (1989), no. 2, 307-340. · Zbl 0691.46011 [10] C. Heil and D. Walnut, Continuous and discrete wavelet transforms , SIAM Rev. 31 (1989), no. 4, 628-666. JSTOR: · Zbl 0683.42031 [11] H. Helson, Lectures on Invariant Subspaces , Academic Press, New York, 1964. · Zbl 0119.11303 [12] A. J. E. M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames , J. Fourier Anal. Appl. 1 (1995), no. 4, 403-436. · Zbl 0887.42028 [13] J. Ramanathan and T. Steger, Incompleteness of sparse coherent states , Appl. Comput. Harmon. Anal. 2 (1995), no. 2, 148-153. · Zbl 0855.42024 [14] M. Rieffel, von Neumann algebras associated with pairs of lattices in Lie groups , Math. Ann. 257 (1981), no. 4, 403-418. · Zbl 0486.22004 [15] A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of $$L_ 2(\mathbf R^ d)$$ , Canad. J. Math. 47 (1995), no. 5, 1051-1094, FTP site anonymous@stolp.cs.wisc.edu, file name frame1.ps. · Zbl 0838.42016 [16] A. Ron and Z. Shen, Frames and stable bases for subspaces of $$L_2(\mathbbR^d)$$: The duality of Weyl-Heisenberg sets , Proceedings of the Lanczos International Centenary Conference, Raleigh, N.C., 1993 ed. D. Brown, et al., SIAM, Philadelphia, 1994, pp. 422-425. [17] A. Ron and Z. Shen, Affine system in $$L_2(\mathbbR^d)$$: The analysis of the analysis operator , to appear in J. Funct. Anal.; FTP site anonymous@stolp.cs.wisc.edu, file name affine.ps. · Zbl 0891.42018 [18] R. Tolimieri and R. S. Orr, Characterization of Weyl-Heisenberg frames via Poisson summation relationships , Proc. ICASSP 92-4 (1992), 277-280. [19] R. Tolimieri and R. S. Orr, Poisson summation, the ambiguity function, and the theory of Weyl-Heisenberg frames , J. Fourier Anal. Appl. 1 (1995), no. 3, 233-247. · Zbl 0885.94008 [20] J. Wexler and S. Raz, Discrete Gabor expansions , Signal Processing 21 (1990), 207-220. [21] M. Zibulski and Y. Y. Zeevi, Matrix algebra approach to Gabor-scheme analysis , Technion Israel Inst. of Tech., vol. 856, EE Pub., September 1992, M. Zibuslski and Y. Y. Zeevi, “Gabor representation with oversampling”, in Conf. on Visual Communication and Image Processing, Proc. SPIE 1818 September 1992, 976-984.
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