Rapidly Growing Fourier Integrals

1. THE RIEMANN–LEBESGUE LEMMA. In its usual form, the Riemann– Lebesgue Lemma reads as follows: If f ∈ L1 and f̂ (s) = ∫∞ −∞ eisx f (x) dx is its Fourier transform, then f̂ (s) exists and is finite for each s ∈ R and f̂ (s) → 0 as |s| → ∞ (s ∈ R). This result encompasses Fourier sine and cosine transforms as well as Fourier series coefficients for functions periodic on finite intervals. When the integral is allowed to converge conditionally, the asserted asymptotic behaviour can fail dramatically. In fact, we show that for each sequence an ↑ ∞ we can find a continuous function f such that f̂ (s) exists for each s ∈ R and f̂ (n) ≥ an for all integers n ≥ 1. We also work out the asymptotics of a class of Fourier integrals that can have arbitrarily large polynomial growth. Our main tool is the principle of stationary phase. The conditionally convergent integrals we consider in this paper can be thought of as Henstock integrals [1] or as improper Riemann integrals. Two examples of conditionally convergent Fourier transforms that do not tend to zero at infinity can be obtained from [3, 3.691]: