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

We show a pointwise estimate for the Fourier transform on the line 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. It is a classical result of Dirichlet that if f is a function of bounded variation on the circle, then the Fourier coefficients, f̂(n), are O(1/n) (and therefore the Fourier series of f converges) [8, p. 57]. We present here an inequality that implies a similar result, but for the Fourier transform on the line. Each time a real 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 the function crests. This paper consists of one theorem and two applications of that theorem. The first application is a new result for the problem of finding nonnegative weights u and v such that the Fourier transform is a bounded operator mapping 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 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 > 0 : |{t : |f(t)| > α}| ≤ x}, where | · | represents the Lebesgue measure 2000 Mathematics Subject Classification. Primary 42A38; Secondary 65T99.