The Sharp Remez-Type Inequality for Even Trigonometric Polynomials on the Period

Abstract

We prove that max t∈ [−π,π] |Q(t)| ≤ T 2n (sec(s/4)) = 1 2 ((sec(s/4) + tan(s/4)) 2n + (sec(s/4) − tan(s/4)) 2n ) for every even trigonometric polynomial Q of degree at most n with complex coefficients satisfyingwhere m(A) denotes the Lebesgue measure of a measurable set A ⊂ R and T 2n is the Chebysev polynomial of degree 2n on [−1, 1] defined by T 2n (cos t) = cos(2nt) for t ∈ R. This inequality is sharp. We also prove that max t∈[−π,π] |Q(t)| ≤ T 2n (sec(s/2)) = 1 2 ((sec(s/2) + tan(s/2)) 2n + (sec(s/2) − tan(s/2)) 2n )for every trigonometric polynomial Q of degree at most n with complex coefficients satisfying ForewordI started my Ph.D. program in June, 1987, at The Ohio State University, as a student of Paul Nevai. In August, 1987, Paul Nevai moved from Columbus, Ohio, to Columbia, South Carolina, to spend the school year 1987-88 at the University of South Carolina, and each of his students followed him. This is how I met Mr. (Yingkang) Hu, who was a student of Ron DeVore at that time. It was a somewhat short but very exciting time for me at the University of South Carolina. Not only had I passed each of my necessary exams in my Ph.D. program but I had the possibility to learn from and interact with some of the very best researchers in approximation theory. My office neighbors were Paul Nevai, Ron DeVore, and George Lorenz. George Lorentz happened to be Ron DeVore's Key words and phrases. trigonometric polynomials, Remez-type inequalities, geometry of polynomials.

Sharp Remez Inequality

“…Theorem 1.1 will follow from the corresponding solution of Problem B for even-degree polynomials. For the partial case of even trigonometric polynomials, i.e., having the form P n (cos t), see also the recent preprint [10]. We point out that we solve Problem B for all integer n. Now we can state our main result.…”