Polynomial properties (Q1826051)

The author discusses the system of polynomials defined by the recurrence relation \(R_{n+1}(x)=2xR_ n(x)-R_{n-1}(x)\) for \(n\geq \ell\) with initial conditions \(Re(x)=G(x),\quad R_{\ell -1}(x)=G_{\ell -1}(x),\) where \(G_{\ell}(x)\), \(G_{\ell -1}(x)\) are polynomials of degree \(\ell\) and \(\ell -1\), respectively, with real coefficients such that all their roots are real and belong to the segment [-1,1] and are mutually interleaved. Special emphasis is put on those systems which are defined in such way that their polynomials deviate the least from zero on the segment -1\(\leq x\leq 1\). We will call them CMBS-systems. The author presents new theorems with necessary and sufficient conditions for the aim that the polynomial system constructed by the recurrence formula will be a CMBS system. He also considers the important application of CMBS systems to the theory of tripartaite iteration methods in a Hilbert space.