Thin interpolating sequences in the disk

Using a kind of summability procedure we give, within this survey, a new proof of a result by Sundberg and Wolff on large perturbations of thin interpolating sequences and present a detailed proof of the statement of Dyakonov and Nicolau that asymptotic interpolation problems in H^∞ can be solved by thin Blaschke products. Dedicated to Professor Heinz K?nig on the occasion of his 80^th birthday     

Finite unions of interpolation sequences

A unified and relatively simple proof is given for some well-known results involving finite unions of uniformly separated sequences.

     

On parametrization of interpolating curves

In parametric curve interpolation there is given a sequence of data points and corresponding parameter values (nodes), and we want to find a parametric curve that passes through data points at the associated parameter values. We consider those interpolating curves that are described by the combination of control points and blending functions. We study paths of control points and points of the interpolating curve obtained by the alteration of one node. We show geometric properties of quadratic Bézier interpolating curves with uniform and centripetal parameterizations. Finally, we propose geometric methods for the interactive modification and specification of nodes for interpolating Bézier curves.     

Rational interpolation through the optimal attachment of poles to the interpolating polynomial

After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the norms of interpolating operators

In this paper we estimate the norms of linear interpolating operators from the space of continuous functions onto polynomials. The estimate eliminates the gap between classical results of Faber and Bernstein. It also provides an affirmative answer to a question recently raised by J. Szabados.     

Complemented invariant subspaces and interpolation sequences

It is shown that the invariant subspace of the Bergman space of the unit disc, generated by a finite union of Hardy interpolation sequences, is complemented in .

     

On growth of norms of Newton interpolating operators

We consider the problem of growth of the sequence of Lebesgue constants corresponding to the Newton interpolation and estimate the growth of this sequence in the case of a nested family of Chebyshev’s points.     

On simultaneous approximation by lagrange interpolating polynomials

This paper considers to replace △_m(x)=(1-x~2)~2(1/2)/n +1/n~2 in the following result for simultaneousLagrange interpolating approximation with (1-x~2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then|f~(k)(x)-P_~(k)(f,x)|=O(1)△_(n)~(a-k)(x)ω(f~(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q,where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_nU Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.     

Maximality properties for isometric interpolating sequences and sequences of trivial points in the spectrum of H∞

Let (x_n) be an isometric interpolating sequence or a sequence of trivial points in the spectrum of H^∞. It is shown that either every cluster point of that sequence has a maximal support set or there exists y ∈ M(H^∞+C) such that the support of x_n is contained in the support of y for infinitely many n. Similar results for Gleason parts are obtained, too. We also investigate the H^∞‐convex hulls of countable unions of support sets and show that whenever supp x ? supp y and x /∈ , then the H^∞‐convex hull of supp x does not meet . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On polyharmonic interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C^∞ or analytic functions in the ball BR. We prove two main results on the interpolation of C^∞ or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.