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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, Roma 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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References

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, 55 (U.S. Government Printing Office, Washington, DC, 1972).Google Scholar
Boas, R. P. Jr., ‘The integrability class of the sine transform of a monotonic function’, Studia Math. 44 (1972), 365–369.CrossRefGoogle Scholar
Boas, R. P. Jr., and Kac, M., ‘Inequalities for Fourier transforms of positive functions’, Duke Math. J. 12 (1945), 189–206.Google Scholar
Bochner, S., Vorlesung über Fouriersche Integrale (Leipzig Verlag, Leipzig, 1932), Annals of Mathematics Studies, 42 (Princeton University Press, Princeton, NJ, 1959); English translation: S. Bochner, Lectures on Fourier Integrals.Google Scholar
Corrington, M. S., ‘Applications of the complex exponential integral’, Math. Comp. 15 (1961), 1–6.CrossRefGoogle Scholar
Giraud, B. G. and Peschanski, R., ‘On positive functions with positive Fourier transforms’, Acta Phys. Polon. B 37(2) (2006), 331–346.Google Scholar
Giraud, B. G. and Peschanski, R., ‘On the positivity of Fourier transforms’, Preprint, 2014, arXiv:1405.3155v1 [math-ph].Google Scholar
Giraud, B. G. and Peschanski, R., ‘From “Dirac combs” to Fourier-positivity’, Acta Phys. Polon. B 47(4) (2016), 1075–1100.CrossRefGoogle Scholar
Giraud, B. G. and Peschanski, R., ‘Fourier-positivity constraints on QCD dipole models’, Phys. Lett. B 760 (2016), 26–30.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, $7$ th edn (eds. Jeffrey, A. and Zwillinger, D.) (Academic Press–Elsevier, Amsterdam, 2007).Google Scholar
Kreyszig, E., Kreyszig, H. and Norminton, E. J., Advanced Engineering Mathematics, 10th edn (Wiley, Hoboken, NJ, 2011).Google Scholar
NIST, Digital Library of Mathematical Functions, (eds. Olver, F. W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R., Saunders, B. V., Cohl, H. S. and McClain, M. A.), Release 1.2.2 (15 September 2024), https://dlmf.nist.gov/.Google Scholar
Olver, F. W. J., Asymptotics and Special Functions (Academic Press, New York, 1974); reprint A.K. Peters/CRC Press, Wellesley, MA, 1997.Google Scholar
Temme, N. M., ‘Error functions, Dawson’s and Fresnel integrals’, in: NIST Handbook of Mathematical Functions (eds. Olver, F. W. J., Lozier, D. M., Boisvert, R. F. and Clark, C. W.) (Cambridge University Press, Cambridge, 2010); Ch. 7.Google Scholar
Tuck, E. O., ‘On positivity of Fourier transforms’, Bull. Aust. Math. Soc. 74 (2006), 133–138.CrossRefGoogle Scholar
Weisstein, E. W., ‘Dawson’s integral’, MathWorld, a Wolfram resource. https://mathworld.wolfram.com/DawsonsIntegral.html.Google Scholar
Weisstein, E. W., ‘Exponential integral’, MathWorld, a Wolfram resource. https://mathworld.wolfram.com/ExponentialIntegral.html.Google Scholar
Wikipedia, ‘Dawson’s function’, https://en.wikipedia.org/wiki/Dawson_function.Google Scholar
Wolfram, Wolfram Mathematica (Wolfram Research, Champaign, IL, 2024). https://www.wolfram.com/mathematica.Google Scholar
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