Coinvariant Subspaces of the Shift Operator and Interpolation

Let Hp(m), 0 < p ^ oo, be the Hardy spaces on a quotient K of the Bohr group. In this paper we completely determine the isometries of Hp{m), p Φ 2, onto itself. Our result is a generalization of a recent work of Muhly who determined the isometries of Hp(m) onto itself under the assumption that the dual group of K is countable, and it may be regarded as a partial answer to a question posed by Muhly. 1* Introduction* Many results have been obtained concerning isometries of Hardy spaces in the theory of uniform algebras. The most fundamental result in this direction is due to de Leeuw, Rudin, and Wermer [2], which states that an automorphism of the classical Hardy space H°°{T) is induced via composition with the unit circle T of a fractional linear transformation of the unit disc onto itself. Their work was carried on independent of Nagasawa [13], who also described the isometries of ίί oo(Tr) onto itself. On the other hand, Arens [1] completely determined the automorphisms of the uniform algebra of analytic functions on a compact abelian group K whose dual group Γ is archimedean ordered (cf. [11]). This result was extended by Muhly [11] to the uniform algebra of analytic functions induced by a flow which has no periodic orbits. Moreover Muhly [12] has recently given, among other things, the following interesting generalization of this result of Arens to the case of isometries of Hardy spaces Hp(m), p Φ 2, on K: Under the assumption that Γ is countable, every isometry of Hp(m), p Φ 2, is induced via composition with an affine map of K such that the adjoint of the additive factor of this map preserves the order of Γ. The purpose of this paper is to remove the assumption on Γ. This result provide a partial positive answer to the following question posed by Muhly in [12; §5]: Is it possible to describe the isometries of ergodic Hardy spaces

In this chapter we introduce shift-invariant algebras, the main objects of this book. These are uniform algebras on a compact connected group G, consisting of continuous functions on G, whose spectrum is contained in a semigroup S of the dual group Ĝ. If Ĝ is a subgroup of ℝ, and S ⊂ ℝ+, then the maximal ideal space of the corresponding shift-invariant algebra is the G-disc, or, big disc over G. In this chapter we describe two important models of shift-invariant algebras, namely, by the means of almost periodic functions on ℝ, and by the means of H∞-functions on the unit circle. The set of automorphisms, and the peak groups of shift-invariant algebras are also characterized. Extensions on G-discs and groups of several classical theorems of Complex Analysis, such as Radó’s theorem for null-sets and the Riemann theorem for removable singularities of analytic functions, are stated and proved. It is shown that these extensions hold for some semigroups, while in general they fail. In principle we state all results for general shift-invariant algebras AS, though they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras. Asymptotically almost periodic functions combine the properties of classical almost periodic functions on ℝ, and of continuous functions on ℝ that vanish at infinity.

Let A be a uniform algebra and M(A) the maximal ideal space of A. A sequence { an }n in M(A) is called .e1-interpolating if for every sequence (an) in f1 there exists a function f in A such that f (an ) = an for all n. In this paper, an f1-interpolating sequence is studied for an arbitrary uniform algebra. For some special uniform algebras, an f1-interpolating sequence is equivalent to an £&lt;)()-interpolating sequence which is familiar for us. However, in general these two interpolating sequences may be different from each other.

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 = ∞.

This thesis is divided into two parts. In the first part, we consider an endomorphism \\alpha: A \\to A, with A being a commutative Banach algebra with unity, and X the spectrum of A. We ensure the existence of a unique partial dynamical system \\varphi: \\Delta \\subseteq X \\to X associated with the endomorphism. By adding the condition that A is a uniform algebra, we obtain an expression for the spectral radius of the weighted endomorphisms a\\alpha, a \\in A, which depends on the invariant or ergodic probabilities measures of the partial dynamical system (X, \\varphi). Moreover, this spectral radius will coincide with the spectral radius of the weighted shift operators aT, where \\alpha(a) = TaS for some operator S. In the second part of the project, for the shift operator \\sigma: \\Delta \\to X_A, where X_A is a generalized Markov shift space, we identify a *-endomorphism \\alpha: \\mathcal_A \\to \\mathcal_A associated with (X_A, \\sigma), where \\mathcal_A is the algebra whose spectrum is X_A. We give examples under conditions that guarantee the continuous extension of the shift operator and other examples where we show the impossibility of extending the shift operator continuously over the entire space. For the particular case of the renewal shift, we characterize every element of the algebra and provide an explicit formula to calculate r(a\\alpha) for all a \\in \\mathcal_A.

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

The starting point is the autocorrelation function of the signal. We assume the latter to be the sum of a random component (noise) plus d trigonometric functions with incommensurate frequencies. The time is descretized and the signal has a finite length N. The autocorrelation is thus itself a set of N random numbers Ci, i = 0,…(N - 1); - periodic boundary conditions are assumed. Considered as Fourier coefficients, the Ci define a random measure dμ with support on the unit. One can then build by a recurrence relation the family of polynomials orthogonal with respect to this measure. These Szego polynomials are simply related to the transfer functions of a family of Wierner-Levinson filters. The n complex random zeros of the random orthogonal polynomial of degree n are the focus point of the method, (i) all the zeros are within the unit circle [1]. When n and N increase, they all have their mean distance D to the unit circle going to zero [2]. (ii) d pairs of complex conjugate zeros can be associated to the d trigonometric components of the signal [3]. In the mean their distance (D) to the unit circle goes to zeros like n –2 [4]. Asymptotically, the modulus of their phases are equal to the frequencies of the d trigonometric components [2]. (iii) the (n – 2d) remaining zeros can be associated to the noise. The bulk of these “noisy zeros” is further away from the unit circle [3]. In the mean, the distance to the latter is decreasing like n – 1 [4]. (iv) if only noise is present the mean positions of the n zeros are regularly spaced on an inner circle [5]. This exactly regular spacing or crystallization is the signature of the noise. It is universal with respect to the probability law of the noise and to the length of the signal, provided lnN.

Shift-type invariant subspaces of contractions

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

Automorphic Forms with Anisotropic Periods on a Unitary Group

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)