Interpolation by Complex Splines

Gibbs–Wilbraham oscillation related to an Hermite interpolation problem on the unit circle

The complex harmonic spline (C.H.S.) is defined by the formula where SM (s) is any complex spline function on the unit circle r = ∂U, with M knots and degree n. In this paper, it is proved that P(z) = f(⊘(z)), where ⊘ is a homeomorphism mapping function and f is an analytic function, the number of zeros of P(z) on U is denoted by N, and we obtain the following result: N = 1/2π Δ r arg P(z) < nM

The results of this chapter can be deduced from the results of Chapter I by means of corresponding substitutions. We prefer an independent analysis of the problems in the circle, as in this way we manage to get rid of certain conditions which we have in Chapter I. These conditions are connected with the possibility of a jump of the distribution function τ(u) at the singular point u = ∞.

In the first part of the paper, it is proved that for 1 < p < ∞ the couple (K , K ∞ ) of coinvariant subspaces of the shift operator on the unit circle is K-closed in the couple (L p (T),L∞ (T)). This property underlies basically all problems of real interpolation for the first couple. Also, a weighted analog of the above statement is established. In the second part it is shown that, given two closed ideals I and J in a uniform algebra such that the complex conjugate of I ∩ J is not included in some of them, the sum I + J is not closed. Though the methods of study in the two parts are quite different, the topics are related by the fact that the question treated in the second part emerged during the work on the first.

Linearized rational interpolation problems at roots of unity play a crucial role in the fast and superfast Toeplitz solvers that we have developed. Our interpolation algorithm is a sequential algorithm in which a matrix polynomial that satisfies already some of the interpolation conditions is updated to satisfy two additional interpolation conditions. In the algorithm that we have used so far, the updating matrix, which is a matrix polynomial of degree one, is constructed in a two-step process that resembles Gaussian elimination. We briefly recall this approach and then consider two other approaches. The first one is a completely new approach based on an updating matrix that is unitary with respect to a discrete inner product that is based on roots of unity. The second one is an application of an algorithm for solving discrete least squares problems on the unit circle, a problem that has linearized rational interpolation at roots of unity as its limiting case. We conduct a number of numerical experiments to compare the three strategies.

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may be written φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)&lt;0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.

In this article uniformly distributed sequences are used to give estimates for a product arising in interpolation problems for analytic functions on the unit circle. This is a continuation of a previous paper [1].

We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

An operator-valued generalization of Tchakaloffʼs theorem

Let B B be a closed algebra lying between H ∞ {H^\infty } and L ∞ {L^\infty } of the unit circle. We define Q A B = H ∞ ∩ B ¯ QA_B = H^\infty \cap \bar {B} , the analytic functions in Q B = B ∩ B ¯ Q_B = B \cap \bar {B} . By work of Chang, Q B {Q_B} is characterized by a vanishing mean oscillation condition. We characterize the sequences of points { z n } \left \{{{z_n}} \right \} in the open unit disc for which the interpolation problem f ( z n ) = λ n , n = 1 , 2 , … f({z_n}) = {\lambda _n}, n = 1, 2,\ldots , is solvable with f ∈ Q B f \in {Q_B} for any bounded sequence of numbers { λ n } \left \{{{\lambda _n}} \right \} . Included as a necessary part of our proof is a study of the algebras Q A B Q{A_B} and Q B {Q_B} .

In 1917/1918 Issai Schur published two papers “On the power series which are bounded in the interior of the unit circle, I and II” in the “Journal fuer Reine und Angewandte Mathematik.” In these two papers the author solved a number of important interpolation problems for classes of analytic functions in the disc. One of them in the famous Schur problem: given complex numbers c0, c1, ... cn find a function $$ f\left( x \right) = \sum\limits_{j = 0}^n {{c_j}{x^j} + \sum\limits_{j = n + 1}^\infty {{f_j}{x^j}} } $$ which is both analytic and bounded by one: |f(x)| ≤ 1, in the disc |x| < 1.

A method of phantom nodes for the case when trigonometric splines are used in the role of theapproximating function is studied [2]. This method involves the fact that a sequence of giveninterpolation nodes are added to an even number of phantom nodes; and the values in these nodes areselected based on estimates of the derivatives, which are measured through the divided differences.Test examples are given; the interpolation error is significantly reduced for these examples.