Kim, H.-J. Fourier and Legendre transform of exponential functions of polynomials. (English) Zbl 0963.42004 Far East J. Math. Sci. (FJMS) 2, No. 5, 703-711 (2000). Summary: J. Chung, D. Kim and S. K. Kim [Math. Res. Lett. 5, No. 5, 629-635 (1998; Zbl 0929.42003)] found a positive number \(c\) such that if \(f(x)= \sum^n_{k=1} a_n x^{2k}\), where \(a_k\geq 0\) and \(a_n> 0\), then \[ |{\mathcal F}(e^{-f})(\omega)|\leq cf^{-1}(1) e^{-bf^*(\omega)}\quad (\omega\in \mathbb{R}) \] for some positive number \(b\) depending only on the degree of \(f\), where \({\mathcal F}\) is the Fourier transform and \(f^*\) is the Legendre transform of \(f\). We improve this inequality. MSC: 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type Keywords:exponential functions of polynomials; Fourier transform; Legendre transform Citations:Zbl 0929.42003 PDFBibTeX XMLCite \textit{H. J. Kim}, Far East J. Math. Sci. (FJMS) 2, No. 5, 703--711 (2000; Zbl 0963.42004)