Polynomials orthogonal on a circular arc

摘要

In this paper complex polynomials {πk}, πk(z)=zk+..., orthogonal with respect to the complex inner product (f,g)=∫π-φφf1(θ)g1(θw1(θ)dθ will be studied. Here φ∈(0,{frcase|1/2}π)and for f(z) the function f1(θ) is defined by f1(θ)=f(-iR+eiθR2+1), R=tan θ. Under suitable integrability conditions on w and assuming the existence of an analytic continuation to the region M+={z∈C:|z+iR|0}, these orthogonal polynomials always exist—in contrast to the case φ=0, the semicircle treated by Gautschi (1989), Gautschi et al. (1987) and Gautschi and Milovanovic´(1986), where an extra condition on Re ∫π0w1(θ)dθ is needed—and they will be expressed in terms of the real polynomials orthogonal on [−1,1] with respect tow(x). This shows the connection between orthogonality on an arc and orthogonality on the spanning chord; on the arc there is no conjugate taken for the second factor.