Riesz-type inequalities on general sets

Sharp Riesz–Bernstein-type inequalities are proven for the derivatives of algebraic polynomials on general subsets of unit circle. The sharp Riesz–Bernstein constant involves the equilibrium density of the set in question.     

Trigonometric series with a generalized monotonicity condition

In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting.In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.     

Chebyshev constants and the inheritance problem

It is proven that any set E consisting of finitely many intervals can be approximated with order 1/n by polynomial inverse images of [-1,1]. This leads to a new proof of the fact that the n-th Chebyshev constant is Kcap(E)ⁿ with some K independent of n. The proof uses properties of monotone systems, in particular, the statement in the so-called inheritance problem.