Mathematical Research Letters

Volume 5, Issue 5, September 1998 pp. 629-635.

Fourier transform of exponential functions and Legendre transform

Authors: Jaeyoung Chung, Dohan Kim and Sung Ki Kim
Author institution: Kunsan National University, and Seoul National University

Summary: 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.

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