Parametrization of spectral surfaces of a class of periodic 5-diagonal matrices

The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices

Explicit min–max polynomials on the disc

AbstractDenote by Πn+m−12≔{∑0≤i+j≤n+m−1ci,jxiyj:ci,j∈R} the space of polynomials of two variables with real coefficients of total degree less than or equal to n+m−1. Let b0,b1,…,bl∈R be given. For n,m∈N,n≥l+1 we look for the polynomial b0xnym+b1xn−1ym+1+⋯+blxn−lym+l+q(x,y),q(x,y)∈Πn+m−12, which has least maximum norm on the disc and call such a polynomial a min–max polynomial. First we introduce the polynomial 2Pn,m(x,y)=xGn−1,m(x,y)+yGn,m−1(x,y)=2xnym+q(x,y) and q(x,y)∈Πn+m−12, where Gn,m(x,y)≔1/2n+m(Un(x)Um(y)+Un−2(x)Um−2(y)), and show that it is a min–max polynomial on the disc. Then we give a sufficient condition on the coefficients bj,j=0,…,l,l fixed, such that for every n,m∈N,n≥l+1, the linear combination ∑ν=0lbνPn−ν,m+ν(x,y) is a min–max polynomial. In fact the more general case, when the coefficients bj and l are allowed to depend on n and m, is considered. So far, up to very special cases, min–max polynomials are known only for xnym,n,m∈N0

Parametrization of spectral surfaces of a class of periodic 5-diagonal matrices

The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices

Parametrization of spectral surfaces of a class of periodic 5-diagonal matrices

The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices