Fourier transform of exponential functions and Legendre transform

We will prove that if $f$ is a polynomial of even degree then the Fourier transform $\Cal F(e^{-f})(\xi)$ can be estimated by $e^{-\epsilon f^*(\xi)}$ where $f^*(\xi)$ is the Legendre transform of $f$ defined by $f^*(\xi) = \sup_x (x\xi -f(x)).$ 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.

Full Text (PDF format)

Published 1 January 1998