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Linear independence of time-frequency translates

Authors:	Christopher Heil Jayakumar Ramanathan Pankaj Topiwala

Institution:	School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160 and The MITRE Corporation, Bedford, Massachusetts 01730 ; Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197 ; The MITRE Corporation, Bedford, Massachusetts 01730

Abstract:	The refinement equation $\varphi (t) = \sum _{k=N_1}^{N_2} c_k \, \varphi (2t-k)$ plays a key role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates $\|a\|^{1/2} \varphi (at-b)$ of $\varphi \in L^2(\mathbf {R})$ , it is natural to ask if there exist similar dependencies among the time-frequency translates $e^{2 \pi i b t} f(t+a)$ of $f \in L^2(\mathbf {R})$ . In other words, what is the effect of replacing the group representation of $L^2(\mathbf {R})$ induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collection $\{(a_k,b_k)\}_{k=1}^N$ , the set of all functions $f \in L^2(\mathbf {R})$ such that $\{e^{2 \pi i b_k t} f(t+a_k)\}_{k=1}^N$ is independent is an open, dense subset of $L^2(\mathbf {R})$ . It is conjectured that this set is all of $L^2(\mathbf {R}) \setminus \{0\}$ .

Keywords:	Affine group frames Gabor analysis Heisenberg group linear independence phase space refinement equations Schroedinger representation time-frequency wavelet analysis

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