A Pointwise Estimate for the Fourier Transform and the Number of Maxima of a Function

We show a pointwise estimate for the Fourier transform involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function. Each time a function changes from increasing to decreasing, we say that the function crests. We show an estimate for the Fourier transform of a function in terms of the number times that the function crests. Under various obvious conditions the number of crests of a function equals the number of local maxima of a function. Hence, there is a connection between the Fourier transform and the number of local maxima of a function. This paper consists of one theorem and two applications of that theorem. The first application is to the problem of finding nonnegative weights u and v such that the Fourier transform maps the weighted Lebesgue space L(v) space into the weighted Lebesgue space L(u). The second application is to finding a lower bound to the number of roots of the derivative of a function by examining its Fourier transform. The functions f we consider are integrable so that their Fourier transform, defined by the formula f̂(z) = ∫ f(x)e dx, exists for all z ∈ R. We provide a precise definition of crests below, but the reader may want to think of them as local maxima for the time being. Theorem. If f ∈ L is nonnegative and #crests(f) ≤ N then |f̂(z)| ≤ Nπ √ 10 ∫ 1/z 0 f∗(x) dx for all z > 0. Here, f is the decreasing rearrangement of f . As usual it is defined by f(x) = inf{α : |{t : |f(t)| > α}| ≤ x}, where | · | represents the Lebesgue measure of a set. In an example below we demonstrate that the N in the theorem is necessary and in a sense sharp. Therefore, we are able to turn our viewpoint and use the contrapositive of the theorem to predict the number of times that the function f will crest. Precisely, the contrapositive is the following. 2000 Mathematics Subject Classification. Primary 42A38; Secondary 65T99.