Volume 2,  Issue 1, 2001

Article 8

AN APPLICATION OF VAN DER CORPUT'S INEQUALITY

KANTHI PERERA
E-Mail: kanthi@engmath.pdn.ac.lk

DEPARTMENT OF ENGINEERING MATHEMATICS
FACULTY OF ENGINEERING
PERADENIYA, SRI LANKA

Received 13 July, 2000; accepted 31 October 2000.
Communicated by: A.M. Fink

ABSTRACT.  In this note we give a short and elegant proof of the result $ \sum_{t=1}^n e^{\imath(\omega t+\alpha t^2)} = o(n) $ for $ \alpha$ 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^{\beta} e^{\imath(\omega t+\alpha t^{2})} = o(n^{\beta+1}) $, where $ \beta$ is a nonnegative constant.
Key words:
Van der Corput's inequality, Hardy and Littlewood.

2000 Mathematics Subject Classification:
42A05.

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