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  Volume 2, Issue 1, Article 8
An Application of Van der Corput's Inequality

    Authors: Kanthi Perera,  
    Keywords: Van der Corput's inequality, Hardy and Littlewood  
    Date Received: 13/07/00  
    Date Accepted: 31/10/00  
    Subject Codes:


    Editors: A. M. Fink,  

In this note we give a short and elegant proof of the result $ sum_{t=1}^n e^{imath(omega t+lpha t^2)} = o(n) $ for $ lpha$ not a rational multiple of $ pi$, uniformly in $ omega$. This was first proved by Hardy and Littlewood, in 1938. The main ingredient of our proof is Van der Corput's inequality. We then generalize this to obtain $ sum_{t=1}^n t^{eta} e^{imath(omega t+lpha t^{2})} = o(n^{eta+1}) $, where $ eta$ is a nonnegative constant.

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