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Zero‐Mean Cosine Polynomials which are Non‐Negative for as Long as Possible
Author(s) -
Gilbert A. D.,
Smyth C. J.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001216
Subject(s) - mathematics , zero (linguistics) , interval (graph theory) , trigonometric functions , discrete orthogonal polynomials , classical orthogonal polynomials , orthogonal polynomials , polynomial , chebyshev polynomials , diophantine equation , difference polynomials , combinatorics , discrete mathematics , mathematical analysis , philosophy , linguistics , geometry
For a given integer n , all zero‐mean cosine polynomials of order at most n which are non‐negative on [0,( n /( n +1))π] are found, and it is shown that this is the longest interval [0,θ] on which such cosine polynomials exist. Also, the longest interval [0,θ] 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 [θ,π] 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.