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Fourier and Legendre transform of exponential functions of polynomials. (English) Zbl 0963.42004

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

Citations:

Zbl 0929.42003
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