Fourier transform of exponential functions and Legendre transform

Abstract

Abstract. We will prove that if f is a polynomial of even degree then the Fourier transform F (e −f )(ξ) can be estimated by e − f * (ξ) where f * (ξ) is the Legendre). This result was previously proved by H. Kang [K] for a case of a convex polynomial which is a finite sum of monomials of even order with positive coefficients. Our result is the most general one for the polynomial f (x) since the convexity condition is not imposed and e −f (x) belongs to the space L 1 if and only if f (x) is a polynomial of even degree with the coefficient of the highest degree a 2m > 0. Also, we will make a more precise estimate of constants.

Instanton Expansions and Phase Transitions

“…We Let us assume that h(p) = 2m k=0 h k p k is a polynomial 25 of even degree, with h 2m > 0 and m ∈ Z + . Then, a theorem by [125] C. Furthermore, using the results of [124], we can further relax this assumption to convex functions h(p) that are analytic within a strip | Im p | < c for c > 0.…”

A proof of hypoellipticity for Kohn’s operator via FBI

“…Actually p 2 (y, ζ 2 ) ∼ y 2kp 2 (y, ζ 2 ). On the integral contour Γ, using the same technique of [4] and [5], we have…”

Upper bounds for Fourier transforms of exponential functions

“…Again, for m = 1 the function F (ω) is equal to its bound √ πe − ω 2 4 in formula (30). Note that in [2] the cruder bound…”

“…To that effect, promising results were obtained in [1,2], where judicious use of the Legendre-Fenchel transform [3] led to meaningful upper bounds. Bounds for Fourier transforms of even more complex exponential functions, the so-called rational exponential integrals [4], where the exponent is a rational function, are still more difficult to obtain.…”

“…where the function S(·) 1 is given by In this paper we modify and generalize the approach adopted in [1,2] in order to obtain significant upper bounds for Fourier integrals under quite general and simple conditions. Several examples illustrating the technique are given.…”