Equivalent norms in some spaces of analytic functions and the uncertainty principle

Abstract

The main object of this work is to describe such weight functions w(t) that for all elements f ∈ Lp,Ω the estimate ‖wf‖p ≥ K(Ω)‖f‖p is valid with a constant K(Ω), which does not depend on f and it grows to infinity when the domain Ω shrinks, i.e. deforms into a lower dimensional convex set Ω∞. In one-dimensional case means that K(σ) := K(Ωσ) → ∞ as σ → 0. It should be noted that in the framework of the signal transmission problem such estimates describe a signal’s behavior under the influence of detection and amplification. This work contains some results of the above-mentioned type which I presented at the Banach Centre in the Summer of 1994. Some of these results had been published earlier, some are new ones. Introduction. Uncertainty principle in Fourier analysis asserts that the more a function f is concentrated the more its Fourier transform F will be spread out. The corresponding nontrivial relations between f and F admit adequate physical interpretations, for instance in the framework of the signal transmission problem, in which the Fourier transform F (ξ) of a signal f(t) is interpreted as a bandwidth. From the physical point of view it is very natural to consider signals of f(t) with compact supported bandwidths F (ξ). Then the function f(t) itself can be extended into the complex space C as an entire function of exponential type. And this is exactly the class of functions we deal with in the course of the paper. More exactly, let f be a function on R and F its Fourier transform defined by F (ξ) = (2π)−n/2 ∫ f(t)e−i dt where t = (t1, t2, . . . , tn), ξ = (ξ1, ξ2, . . . , ξn) are points of R, = t1ξ1+. . .+tnξn. For 1 ≤ p ≤ ∞ we denote by ‖f‖p the Lp-norm of a function f . Let Ω ⊂ R be an arbitrary bounded domain and let 1 ≤ p ≤ ∞. We denote by Lp,Ω the space of all functions f such that the norm ‖f‖p is finite, and the Fourier transforms F of f are supported in Ω. Such functions f vanish at infinity in R and can be extended into the 1991 Mathematics Subject Classification: Primary 30D10; Secondary 81B10. The paper is in final form and no version of it will be published elsewhere.