Remark on the formula by Rakhmanov and Steklov's conjecture

TL;DR

This paper revisits Rakhmanov's negative solution to Steklov's conjecture, demonstrating how orthogonal polynomials can be derived using a recent method for establishing sharp lower bounds.

Contribution

It connects Rakhmanov's formula with a new method for lower bounds, providing insight into the polynomial construction related to Steklov's conjecture.

Findings

Rakhmanov's formula can be derived using the recent method for lower bounds

The approach offers a new perspective on orthogonal polynomial construction

Clarifies the relationship between Rakhmanov's proof and modern techniques

Abstract

The conjecture by Steklov was solved negatively by Rakhmanov in 1979. His original proof was based on the formula for orthogonal polynomial obtained by adding point masses to the measure of orthogonality. In this note, we show how this polynomial can be obtained by applying the method developed recently for proving the sharp lower bounds for the problem by Steklov.