Gabor orthogonal bases and convexity

Gabor orthogonal bases and convexity, Discrete Analysis 2018:19, 11 pp.

A fundamental way of understanding a function \(f\) defined on \(\mathbb R^d\) is to expand it in terms of a basis with nice properties. Typically, one assumes that \(f\in L_2(\mathbb R^d)\), and then it becomes natural to look for orthonormal bases with properties such as interesting symmetries. For example, wavelet bases, which play a very important role in signal processing, are orthonormal and consist of translates and dilates of a single function.

One class of “good” bases that has been studied is the class of Gabor orthogonal bases. Here the idea is to take a countable set of functions of the form \(g_{a,b}(x)=g(x-a)\exp(-2\pi ix.b)\), where \(g\) is a fixed function in \(L_2(\mathbb R^d)\) and \(a,b\) are elements of \(\mathbb R^d\). One way to achieve this is to find a measurable set \(K\) that tiles \(\mathbb R^d\) with the property that there is an orthonormal basis of \(L_2(K)\) that consists of functions of the form \(\chi_K(x)\exp(-2\pi ix.b)\). Then we can obtain a similar orthonormal basis for \(L_2(K+t)\) for each translate \(t\) in the tiling, and combining these bases one obtains a basis for the whole of \(L_2(\mathbb R^d)\), for the simple reason that there is no interaction between the bases for the different translates. It was conjectured by Fuglede in 1974 that a set \(K\) admits an orthonormal basis of exponentials as above if and only if it tiles \(\mathbb R^d\), but this was shown not to be true by Tao in 2003. Understanding which sets are “spectral” in this way is still an active area of research.

In general, our understanding of which functions \(g\) can be used to form Gabor orthogonal bases is quite limited. In particular, it seems to be hard to prove negative results. In this paper, it is shown that a certain class of functions cannot be used: if \(K\) is a convex body with a smooth boundary with Gaussian curvature that does not vanish anywhere, then provided that \(d\ne 1 \mod 4\), the characteristic function of \(K\) cannot be used as the function \(g\) in a Gabor orthogonal basis. Of course, there is no chance of tiling \(\mathbb R^d\) with such a set \(K\), but that on its own does not rule out some kind of complicated interaction between neighbouring “local” bases.

The condition that the Gaussian curvature does not vanish everywhere is a strong one, but the authors believe that it should not be too hard to modify the proof to do without it, since there will be plenty of points with non-vanishing curvature even if not all of them have it. The condition that \(d\ne 1\mod 4\) is more mysterious – late on in the proof, a contradiction is obtained, which fails to be a contradiction if \(d=1\mod 4\), but that appears to be an artefact of the proof rather than a serious indication that the result is different in that case.

More or less all the obvious questions one might want to ask about Gabor orthogonal bases are still open. For example, it is not known whether there is a non-spectral set \(K\) such that \(\chi_K\) will work. In this case, one would not be able to find an orthonormal basis of exponentials for \(L_2(K)\), so one would have to combine non-spanning orthonormal bases of exponentials for overlapping translates of \(K\). A particularly interesting case that the authors mention is that of a triangle in \(\mathbb R^2\).