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

Published:2011-04-06
Printed: Dec 2012
• Ryan Berndt,
Yale University and Otterbein College, Otterbein College, Westerville, Ohio 43081
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Abstract

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
 MSC Classifications: 42A38 - Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 65T99 - None of the above, but in this section