Sharpening of Hilbert’s lemniscate theorem

In this paper, we extend Hilbert’s lemniscate theorem to touching systems of curves. The result allows finding sharp constants in Bernstein type inequalities. Supported by OTKA TS44782. Supported by NSF grant DMS-0097484 and by OTKA T/034323, TS44782.     

Summability in the classes Hω

Mathematical Notes -     

Extensions of Szegö's theory of orthogonal polynomials,II

Let {? _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior of? _n (dμ) when logμ'∈L ¹. In what follows we will discuss the asymptotic behavior of the ratio φ_n(dμ ₁)/φ_n(dμ ₂) off the unit circle in casedμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=g dμ ₁ whereg≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0, for the monic polynomials Φ _n . The consequences for orthogonal polynomials on the real line are also discussed.     

Obituary Ferenc Móricz (1939–2021)

Analysis Mathematica -