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A CRITERION FOR ENSURING POSITIVITY OF FOURIER TRANSFORMS

Part of: Harmonic analysis in one variable

Published online by Cambridge University Press:  20 August 2025

RENATO SPIGLER Open the ORCID record for RENATO SPIGLER [Opens in a new window]
RENATO SPIGLER*
Affiliation:
Department of Mathematics and Physics, https://ror.org/05vf0dg29Roma Tre University, 1, Largo S. Leonardo Murialdo, 00146 Rome, Italy
*
e-mail: renato.spigler@uniroma3.it
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Abstract

It is shown that the Fourier sine transform, $\mathcal{F}_S [f(t)](\omega )$ on $\mathbb {R}_0^+$, of any given real-valued function $f(t)$ that does not vanish at $t=0$ or has a nonvanishing even-order derivative at $t=0$, has a definite sign at least for $\omega> \omega _0$, where $\omega _0$ can be estimated. Similarly, the cosine transform, $\mathcal{F}_C [f(t)](\omega )$, of functions with a nonvanishing odd-order derivative at zero also has a definite sign for sufficiently large $\omega $. Several examples are given.

Keywords

Fourier transformspositivity of Fourier sine and cosine transforms

MSC classification

Primary: 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 11
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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