Interpolation Problems in the Unit Circle

The dependence of the location of singular points of orthorhombic (ORT) media on the stiffness coefficients , and phase velocity at the singularity point is studied under the assumption that are larger than and . In this case, singular points appear only at the intersection of slowness surfaces of S1- and S2-waves. To simplify the presentation of the results, the values , are fixed and changed within the limits at which the stiffness matrix remains positive definite. We define the parameters, , , , which result in 0, 1, or 2 singular points in the symmetry planes of the ORT medium. The types of these singular points and their location on the unit circle are described. It is shown that, by selecting parameters , any singular point in the symmetry plane 13 can be combined with the limiting position of the singularity point in-between the symmetry planes, or this point can be included in the singular curve of the degenerate ORT medium. We derive equations for the semi-axes of an ellipse of conical refraction, which is the image of a singular point from plane 13 in the group domain. Conditions for degeneration of the ellipse of conical refraction into a segment or a point are defined. It is shown that there exists only one ORT medium with a fixed phase velocity of S1- and S2-waves in a given singular direction n. If we present the ORT media as projections of these vectors n onto the plane 12 and mark the value of the Poincaré index of the S1- or S2-wave at the point n, we get 2 regions with indices 1/2 and –1/2 separated by projection of the singular curve in the form of an ellipse or hyperbola. We compute equations for of a degenerate ORT medium in terms of the values, , and velocity of S1-, S2-waves on a singular curve. The singular curve is defined by the intersection of a unit sphere with an elliptic cone. It is proved that a degenerate ORT medium for or is, respectively, a transversally isotropic medium with a vertical or horizontal axis of symmetry. The results are illustrated in several examples.

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.

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

We obtain new explicit forms of the Schwarz integral in the unit circle. By means of superpositions of logarithms we establish criteria for the appearance of singular boundary points and circles in inverse boundary-value problems.

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 this paper we solve the problem of interpolation by certain class of cardinal complex splines.This solution is used to complete the study of cardinal trigonometric splines started in [10] and also to give shorter proofs and to complete the results found for the interpolation problem by complex splines over the unit circle by I.

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)

Constrained zones at singular points of inclusion contours

To establish the theorem we first note that the condition (2)' (a) is sufficient to insure the convergence of (1) in the wider sense, and hence to a meromorphic function, for all \z | <^1.

For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations of Wiegmann and Zabrodin. When the magnetic flux per plaquette is $\frac{1}{Q}$ with $Q$ an odd integer, distribution of the roots of the Bethe ansatz equation is uniform except at two points on the unit circle in the complex plane. For the semiclassical limit $Q\ensuremath{\rightarrow}\ensuremath{\infty}$, the wave function is ${|\ensuremath{\psi}(x)|}^{2}=(\frac{2}{sin \ensuremath{\pi}x})$, which is critical and unnormalizable. For the golden-mean flux, the distribution of roots has exact self-similarity and the distribution function is nowhere differentiable. The corresponding wave function also shows a clear self-similar structure.

An asymptotic formula is found for the derivatives of orthogonal polynomials on the unit circle. The condition on the distribution function is essentially local and the result is stronger than those known before.