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van der Corput's Inequality

Let y_n be a complex number for 1<=n<=N and let y_n=0 if n<1 or n>N. Then

|sum_(n=1)^Ny_n|^2<=(N+H)/(H+1)sum_(n=1)^N|y_n|^2+(2(N+H))/(H+1)sum_(h=1)^H(1-h/(H+1))|sum_(n=1)^(N-h)y_(n+h)y^__n|

(Montgomery 2001).

See also

Weyl Sum

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References

Montgomery, H. L. "Harmonic Analysis as Found in Analytic Number Theory." In Twentieth Century Harmonic Analysis--A Celebration. Proceedings of the NATO Advanced Study Institute Held in Il Ciocco, July 2-15, 2000 (Ed. J. S. Byrnes). Dordrecht, Netherlands: Kluwer, pp. 271-293, 2001. http://www.nato-us.org/analysis2000/papers/montgomery.pdf.

Referenced on Wolfram|Alpha

van der Corput's Inequality

Cite this as:

Weisstein, Eric W. "van der Corput's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/vanderCorputsInequality.html

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