Interpolation by Rational Functions with Nodes on the Unit Circle

Abstract

From the Erdős–Turán theorem, it is known that if f is a continuous function on ${\Bbb T} = \left\{ {z:\left\lfloor z \right\rfloor = 1} \right\}$ and L _n (f, z) denotes the unique Laurent polynomial interpolating f at the (2 n + 1)th roots of unity, then $\mathop {\lim }\limits_{n \to \infty } \int_{\Bbb T} {\left| {f\left( z \right)} \right|^2 } \left| {{\text{d}}z} \right| = 0$ Several years later, Walsh and Sharma produced similar result but taking into consideration a function analytic in ${\Bbb D} = \left\{ {z:\left| z \right| < 1} \right\}$ and continuous on ${\Bbb D} \cup {\Bbb T}$ and making use of algebraic interpolating polynomials in the roots of unity.

In this paper, the above results will be generalized in two directions. On the one hand, more general rational functions than polynomials or Laurent polynomials will be used as interpolants and, on the other hand, the interpolation points will be zeros of certain para-orthogonal functions with respect to a given measure on ${\Bbb T}$.