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Journal of the London Mathematical Society 2000 62(2):489-504; doi:10.1112/S0024610700001216
© 2000 by London Mathematical Society
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© The London Mathematical Society

Zero-Mean Cosine Polynomials which are Non-Negative for as Long as Possible

A. D. Gilbert and C. J. Smyth

Department of Mathematics and Statistics, University of Edinburgh James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ tony2{at}maths.ed.ac.uk, chris{at}maths.ed.ac.uk

Received 15 October 1998.

For a given integer n, all zero-mean cosine polynomials of order at most n which are non-negative on [0,(n/(n+1)){pi}] are found, and it is shown that this is the longest interval [0,{theta}] on which such cosine polynomials exist. Also, the longest interval [0,{theta}] on which there is a non-negative zero-mean cosine polynomial with non-negative coefficients is found.

As an immediate consequence of these results, the corresponding problems of the longest intervals [{theta},{pi}] on which there are non-positive cosine polynomials of degree n are solved.

For both of these problems, all extremal polynomials are found. Applications of these polynomials to Diophantine approximation are suggested.


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