An extremum problem for polynomials related to codes and designs (Q1585648)

In this interesting paper, the authors solve an extremum problem for polynomials. Let \(\sigma\) be a positive Borel measure on \([-1,1]\) with associated orthogonal polynomials \(\{P_n\}^\infty_{n=0}\), normalized by the condition \(P_n(1)=1\), \(n\geq 0\). Let \(H_n(\sigma)\) denote the polynomials of degree \(\leq n\) admitting a representation in terms of the orthogonal polynomials, with non-negative coefficients: \(P\in H_n(\sigma) \Leftrightarrow P= \sum^n_{j=1} a_jP_j\) with all \(a_j\geq 0\). In a problem arising from the theory of codes and designs, one seeks to find \(B_n(\sigma): =\sup\{\lambda \in [-1,1]: \exists P\in H_n (\sigma)\) such that \(P\geq 0\) in \([-1,\lambda]\}\). The authors obtain bounds and in some cases identities for \(B_n(\sigma)\). For example, \(B_{2n+1} (\sigma)\leq t_2\) where \(t_2\) is the second largest zero of the orthogonal polynomial of degree \(n\) for the measure \((1+t)d \sigma(t)\). In the case where \(P_iP_j\) belongs to \(H_{i+j} (\sigma)\), one has \(B_{2n} (\sigma)= t_2\), where \(t_2\) is the second largest zero of \(P_{n+1}\). Further results of this type are given. The authors relate their work to partial results of other researchers.