On a Remez-type inequality for trigonometric polynomials

AbstractWe obtain a Remez-type inequality for a trigonometric polynomial Qn of degree at most n with real coefficients ‖Qn‖C((−π,π]≤(1/2)(2/sin(λ/4))2n‖Qn‖C(E),λ∈(0,2π], where E⊆(−π,π] is a measurable set with |E|≥λ. This estimate is asymptotically sharp as λ→0+, that is, for the best constant Cn,R(λ) in this inequality, Cn,R(λ)=(1/2)(8/λ)2n(1+o(1)). We also extend this result to polynomials with complex coefficients

Constructive Function Theory on Sets of the Complex Plane through Potential Theory and Geometric Function Theory

This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory

Remez-type inequalities and their applications

AbstractThe Remez inequality gives a sharp uniform bound on [−1, 1] for real algebraic polynomials p of degree at most n if the Lebesgue measure of the subset of [−1, 1], where |;p|; is at most 1, is known. Remez-type inequalities give bounds for classes of functions on a line segment, on a curve or on a region of the complex plane, given that the modulus of the functions is bounded by 1 on some subset of prescribed measure. This paper offers a survey of the extensive recent research on Remez-type inequalities for polynomials, generalized nonnegative polynomials, exponentials of logarithmic potentials and Müntz polynomials. Remez-type inequalities play a central role in proving other important inequalities for the above classes. The paper illustrates the power of Remez-type inequalities by giving a number of applications