Orthogonal polynomials on several intervals via a polynomial mapping

Abstract:

Starting from a sequence $\{ {p_n}(x; {\mu _0})\}$ of orthogonal polynomials with an orthogonality measure ${\mu _0}$ supported on ${E_0} \subset [ - 1, 1]$, we construct a new sequence $\{ {p_n}(x; \mu )\}$ of orthogonal polynomials on $E = {T^{ - 1}}({E_0})$ ($T$ is a polynomial of degree $N$) with an orthogonality measure $\mu$ that is related to ${\mu _0}$. If ${E_0} = [ - 1, 1]$, then $E = {T^{ - 1}}([ - 1, 1])$ will in general consist of $N$ intervals. We give explicit formulas relating $\{ {p_n}(x; \mu )\}$ and $\{ {p_n}(x; {\mu _0})\}$ and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses $T$ to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.