Treffer: A note on hermite-fejér interpolation for the unit circle: A note on Hermite-Fejér interpolation for the unit circle

For every pair \((p,q)\) of integers, where \(p\leq q\) we denote by \(\Lambda_{p,q}\) the linear space of all Laurent polynomials (\(L\)-polynomials) \[ L(z)= \sum_{j=p}^q c_j z^j, \quad c_j\in \mathbb{C}. \] Suppose that \(p(n)\), \(q(n)\) are two nondecreasing sequences of nonnegative integers such that \(p(n)+ q(n)= 2n-1\), \(n= 1,2,\dots\) with \(\lim_{n\to\infty}(n)= \lim_{n\to\infty}q(n)= \infty\), and \(|p(n)-n|\) bounded. In this paper the following result is proved: Theorem. Let \(f\) be a continuous function on \(\mathbb{T}= \{z\in \mathbb{C}: |z|= 1\}\) and let \(L_n\) be the unique \(L\)-polynomial in \(\Lambda_{-p(n),q(n)}\) satisfying \[ L_n(z_k)= f(z_k), \quad L_n'(z_k)= 0, \qquad k=1,2,\dots, n \] where \(z_k\) are the roots of \(z^n+ \lambda_n= 0\), \(\{\lambda_n\}\) being an arbitrary sequence on \(\mathbb{T}\). Then the sequence \(L_n\) converges to \(f\) uniformly on \(\mathbb{T}\). This result is an extension to the unit circle \(T\) of the classical Hermite-Fejér theorem about an approximation on the interval \([-1,1]\).