This analysis reveals that Fourier sine and cosine transforms maintain a definite sign under specific conditions, indicating their applicability in analysis.
It is shown that the Fourier sine transform, FS [f(t)](ω ) on R₀⁺ , 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 ω> ω ₀ , where ω ₀ can be estimated. Similarly, the cosine transform, FC [f(t)](ω ) , of functions with a nonvanishing odd-order derivative at zero also has a definite sign for sufficiently large ω . Several examples are given.
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Renato Spigler (2025) studied this question.
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