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.

Keywords:

Fourier transform, maxima, two weight problem, roots, norm estimates, Dirichlet-Jordan theorem Fourier transform, maxima, two weight problem, roots, norm estimates, Dirichlet-Jordan theorem

MSC Classifications:

42A38, 65T99 show english descriptions Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
None of the above, but in this section 42A38 - Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
65T99 - None of the above, but in this section

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