  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.
