## 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.

### MSC:

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

### Keywords:

interpolation; nodes; Laurent polynomials