Let \(\{P_ n\}_{n=0}^ \infty\) be a system of orthogonal polynomials. R. Lasser observed that if the linearization coefficients of \(\{P_ n\}_{n=0}^ \infty\) are nonnegative then each of the \(P_ n\) is a linear combination of the Tchebyshev polynomials with nonnegative coefficients. The aim of this paper is to give a partial converse to this statement. We also consider the problem of determining when the polynomials \(P_ n\) can be expressed in terms of \(Q_ n\) with nonnegative coefficients, where \(\{Q_ n\}_{n=0}^ \infty\) is another system of orthogonal polynomials. New proofs of well known theorems are given as well as new results and examples are presented.