Bump Functions With Monotone Fourier Transforms Satisfying Decay Bounds

Abstract

The existence of a smooth, compactly supported function with monotone (on the half-line) Fourier transform satisfying two-sided decay bounds is demonstrated.

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“…More interestingly, there are also absolutely continuous distributions with this property. In [13], a function f ∈ C ∞ (R) which is real, nonnegative, symmetric, supported on [−1, 1], not identically equal to zero and such that its (real-valued) Fourier transform f (t) is monotone decreasing for t ≥ 0 (and hence nonnegative) is constructed. After possibly rescaling, any such f is the probability density function of some absolutely continuous random variable, which by Theorem 1.8 is reasonable with respect to cos(x).…”

“…Assume that Ef (a + dX) is well defined and finite for all a ∈ R n , d > 0. Letting, as we do for payoff functions, Ef (a + dX), (13) assume that there exist d 1 < d 2 so that g X,f (d 1 ) < g X,f (d 2 ). Then there exists a payoff function f ′ (hence bounded from above by definition) so that (X, f ′ ) is not reasonable.…”