On Uncertainty Bounds and Growth Estimates for Fractional Fourier Transforms

Gelfand-Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this paper we consider mapping properties of fractional Fourier transforms on Gelfand-Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space. 1 Fractional Fourier transforms on Gelfand-Shilov spaces The uncertainty principle says that a function and its Fourier transform cannot both decay too rapidly. When interpreted as a statement about pointwise decay at infinity, perhaps the most familiar version of this statement is Hardy’s theorem [H]: if |f(x)| ≤ Ce−παx2 while |f̂(ξ)| ≤ C ′e−πβξ2 then: if αβ = 1 then f is necessarily a multiple of the Gaussian e−παx 2 . Consequently, if αβ > 1 then f is the zero function. In other forms of the uncertainty principle it is difficult to identify a single optimizer, but still important to identify a class of near optimizers. An example of this is Beurling’s theorem ([HOR], see also [BDJ]): if ∫ ∫ |f(x)||f̂(ξ)|e2π|xξ| dx dξ < ∞ then f = 0. An important corollary of this quantifies limitations on joint decay in time and frequency in terms of a conjugate pair M and M∗ of Young’s functions. Suppose that m(t) is positive and, for the sake of the present discussion, strictly increasing on (0,∞). Define M(t) = ∫ t 0 m(τ) dτ and M∗(s) = ∫ s 0 m−1(t)dt where m−1 is the inverse function of m. The convex function M is called a Young’s function with dual M∗. One extends M and M∗ to all of R by setting M(x) = M(−x) and M∗(y) = M∗(−y). Standard examples include M(x) = |x|/p for which M∗(y) = |y|p/p′ where p > 1 and p′ = p/(p− 1) is its conjugate exponent. Conjugate Young’s functions satisfy Young’s inequality: |xy| ≤ M(x) + M∗(y), with equality when y = m(x). This yields the following corollary to Beurling’s theorem: if ∫ |f(x)|e dx < ∞ and ∫ |f̂(ξ)|e2πM(ξ) dξ < ∞ then f = 0. In the case M(x) = αx/2 one has M∗(ξ) = ξ/(2α) and the conditions are akin to those of Hardy’s theorem, except that pointwise decay is replaced by convergence of an integral. Hörmander [HOR] cited this case as an illustration of the sharpness of the corollary. He also cited Gelfand-Shilov spaces as illustrations of its sharpness for certain more general M in the sense of the existence of nontrivial elements f of these spaces such that, for a given pair α, β with αβ < 1, one has ∫ |f(x)|e dx < ∞ and ∫ |f̂(ξ)|e2πβM(ξ) dξ < ∞. Such elements are, in a sense, nearly ideally time-frequency localized with respect to the dual pair M,M∗.