Geodesic
Some Results Relating the Behaviour of Fourier Transforms Near the Origin and at Infinity
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 942-950

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It is known that under special conditions, Fourier sine transforms and Fourier cosine transforms behave asymptotically like a power of x, either as x → 0 or as x → ∞ or both. For example (3), where f(x) = x–αφ(x), 0 < α < 1, and φ(x) is of bounded variation in (0, ∞) and Fc(x) is the Fourier cosine transform of f(x). This suggests that other results connecting the behaviour of a function at infinity with the behaviour of its Fourier or Watson transform near the origin might exist. In this paper wre derive various such results. For example, a special case of these results is where f(x) is the Fourier sine transform of g(x). It should be noted that the Fourier inversion formula fails to give f(+0) directly in this case. Some applications of these results to show the relationships between various forms of known summation formulae are given. 
Nasim, C. Some Results Relating the Behaviour of Fourier Transforms Near the Origin and at Infinity. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 942-950. doi: 10.4153/CJM-1969-103-5
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