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Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. (English) Zbl 0849.43003

Wavelets: a tutorial in theory and applications, Wavelet Anal. Appl. 2, 359-397 (1992).
In this paper, the authors study series expansions of functions (signals) with respect to Gabor wavelets, via the group point of view, broadly developed in their previous works and applied here to the (reduced) Heisenberg group \(\mathbb{R}^{2d} \times \mathbb{T} = \mathbb{H}_d\) [J. Funct. Anal. 86, 307-340 (1989; Zbl 0691.46011); Monatsh. Math. 108, 129-148 (1989; Zbl 0713.43004)]. They start with the short-time Fourier transform: \[ S_g f(x,y) = \int_{\mathbb{R}^d} e^{-2 \pi iy \cdot z} \overline g (z - x )f(z)dz = \langle M_y T_x g,f \rangle, \] where \(T_x f(z) = f(z - x)\), \(M_y f(z) = e^{2 \pi iz \cdot x} f(z)\) and \(g\) is an analysing function (a “window”). For \(a > 0\) and for a weight \(m\) (strictly positive, continuous and such that \(m(x + y) \leq c(1 + |x |)^a\), \(x,y \in \mathbb{R}^d)\), they define a modulation space \[ M^m_{p,q} (\mathbb{R}^d) = \left \{f \in {\mathcal S}' (\mathbb{R}^d) : \int_{\mathbb{R}^d} \left( \int_{\mathbb{R}^d} \bigl |S_g f(x,y) \bigr |^p m^p(x,y) dx \right)^{q/p} dy < \infty \right\} \] (with \(1 < p < \infty\), \(1 < q < \infty\), and natural change for \(p = \infty\) or \(q = \infty)\). Then their principal result is the following (Theorem 7, p. 367): For any \(g \in L^2 (\mathbb{R}^d)\) with Wigner distribution \(W_g\) in \(L^1 (\mathbb{R}^d, (1 + |x |)^a dx)\), there exists a neighborhood \(U\) of \((0,0)\) in \(\mathbb{R}^{2d}\) such that, for any \(U\)-dense family \((x_i, y_i)\), \(i \in I\), in \(\mathbb{R}^{2d}\), it is possible to write any \(f \in M^m_{p,q} (\mathbb{R}^d)\) as \(f = \sum c_i Tx_i My_i g\), the family \((c_i)\) belonging to a (well-determined) sequence space.
For the proof they use a “correspondence principle” between \(\mathbb{R}^{2d}\) and \(\mathbb{H}_d\). In fact, the family \(\{\tau T_x M_y : \tau \in \mathbb{C}\), \(|\tau |= 1\), \(x,y \in \mathbb{R}^d\}\) forms a group of unitary operators in \(L^2 (\mathbb{R}^d)\) which is the Schrödinger representation \(\Pi\) of \(\mathbb{H}_d\). Moreover, the coefficients \(V_g f(h) = \langle \Pi (h)g,f \rangle\) are uniquely determined by their values on \(\mathbb{R}^{2d} \times \{1\}\) [cf. R. E. Howe, Bull. Am. Math. Soc., New Ser. 3, 821-843 (1980; Zbl 0442.43002)]. The representation \(\Pi\) is integrable (“a fortiori” square integrable) and one has the “smart” and yet useful reproducing formula \(V_g (f) = V_g (f)* V_g (g)\) (where \(*\) is the convolution group law).
The authors also give stable algorithms (called “adaptive weights methods”). Those algorithms converge unconditionally in modulation spaces and are compared with the frame approach [I. Daubechies, IEEE Trans. Inf. Theory 36, 961-1005 (1990; Zbl 0738.94004)].
At the end of this paper, the authors make some remarks concerning invariance properties of modulation spaces (via the metaplectic group) and the unitary equivalent Bargmann-Fock representation.
For the entire collection see [Zbl 0744.00020].

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
22D10 Unitary representations of locally compact groups
22E25 Nilpotent and solvable Lie groups
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65D15 Algorithms for approximation of functions
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