Some results about interpolation with nodes on the unit circle. (English) Zbl 0971.41003

Since a continuous function can be uniformly approximated by Laurent polynomials of the form \(L(z)= \sum^q_{j=-p} a_jz^j\), \(p,q\) some natural numbers, then it seems natural to study the convergence of sequences of interpolating Laurent polynomials. Let \(\Lambda_{-p,q}\) denote the spaces of Laurent polynomials as given above. The important result of this paper may be presented as follows: Theorem: Let \(T=\{z:|z|=1\}\) and let \(f\) be the given continuous function on \(T\) with the modulus of continuity \(\omega (f,\delta) =O (\delta^p)\) for \(p>{1\over 2}\) and \(\delta\to 0\). Let \(p(n)\), \(q(n)\) be the given sequences of the natural numbers such that \(p(n)+q(n)=n\), \(\lim_{n\to \infty} {p(n)\over n}= \gamma\in (0,1)\). If \(R_n(x)= R_n(f,x)\) is the best interpolant of the function \(f\) in the spaces \(\Lambda_{-p(n),q(n)}\) with the nodes \(x_{j,n}\), \(j=1,2, \dots,n+1\), being the \((n+1)\)th roots of \(w_n\), \(|w_n|=1\), \(n=1,2,\dots, R_n (x_{j,n}) =f(x_{j,n})\), then \(R_n(x)\) is uniformly convergenct on \(T\) to the function \(f(x)\). Some other similar problems are also investigated.


41A10 Approximation by polynomials
30E10 Approximation in the complex plane
41A05 Interpolation in approximation theory