ORTHOGONAL AND EXTREMAL POLYNOMIALS ON SEVERAL INTERVALS

Let a1 < a2 < ... < a2l, E(l) = or j=1l[a2j-1, a2j], H(x) = PI(j=1)2l(x - a(j)) and let rho be a polynomial which has no zero in int(E(l)). In this paper, which is of survey character, first minimal polynomials with respect to the max norm and weight 1/square-root rho, where rho is positive on E(l), are characterised by orthogonality conditions and some new results are proved. Then the connection with polynomials orthogonal with respect to square-root Absolute value of H / Absolute value of rho, where rho has an odd number of zeros in each interval [a2j, a2j+1], j = 1,..., l - 1, is shown and a full description of orthogonal polynomials having periodic respectively asymptotic periodic recurrence coefficients is given. Finally the asymptotic behaviour of polynomials orthogonal on several intervals is discussed.