A pointwise estimate for the Fourier transform and the number of maxima of a function

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Berndt, Ryan
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Abstract
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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.
Comment: 9 pages, fixed typos, extended explanation of an application
Keywords
Mathematics - Classical Analysis and ODEs, Mathematics - Numerical Analysis, 42A38, 65T99
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