THE NORM OF TIME-FREQUENCY AND WAVELET LOCALIZATION OPERATORS

Time-frequency localization operators (with Gaussian window) L_F : L^2(R^d) → L^2(R^d), where F is a weight in R^2d, were introduced in signal processing by I. Daubechies [IEEE Trans. Inform. Theory 34 (1988), pp. 605–612], inaugurating a new, geometric, phase-space perspective. Sharp upper bounds for the norm (and the singular values) of such operators turn out to be a challenging issue with deep applications in signal recovery, quantum physics and the study of uncertainty principles. In this note we provide optimal upper bounds for the operator norm ||L_F||_{L^2→L^2}, assuming F ∈ L^p(R^2d), 1 < p < ∞ or F ∈ L^p(R^2d) ∩ L^∞(R^2d), 1 ≤ p < ∞. It turns out that two regimes arise, depending on whether the quantity ||F||_{L^p}/||F||_{L^∞} is less or greater than a certain critical value. In the first regime the extremal weights F, for which equality occurs in the estimates, are certain Gaussians, whereas in the second regime they are proved to be Gaussians truncated above, degenerating into a multiple of a characteristic function of a ball for p = 1. This phase transition through Gaussians truncated above appears to be a new phenomenon in time-frequency concentration problems. For the analogous problem for wavelet localization operators—where the Cauchy wavelet plays the role of the above Gaussian window—a complete solution is also provided.