Theory Of Hardy Spaces Research Articles

A convergence theorem for spectral factorization

This paper presents a convergence theorem for an iterative method of spectral factorization in the context of multivariate prediction theory. It may be viewed as a constructive proof that the factorization exists, using only the analytic results of Hardy space theory.

Correction to: ``On the distributions of bounded functions in the abstract Hardy space theory and some of their applications''

Correction to: ``On the distributions of bounded functions in the abstract Hardy space theory and some of their applications''

M. Riesz's theorem in the abstract Hardy space theory

M. Riesz's theorem in the abstract Hardy space theory

Fourier analysis on the Heisenberg group

We obtain a usable characterization of the (group) Fourier transform of (H(n)) (Schwartz space on the Heisenberg group). The characterization involves writing elements of [Formula: see text] as asymptotic series in Planck's constant. In the process, we derive a new "discrete" version of spherical harmonics, and elucidate the theory of group contractions. We give an application to Hardy space theory.

On the distributions of bounded functions in the abstract Hardy space theory and some of their applications

On the distributions of bounded functions in the abstract Hardy space theory and some of their applications

Sobolev approximation by a sum of subalgebras on the circle

Let ψ be an orientation-reversing homeomorphism of the unit circle onto itself. We consider approximation in certain Sobolev norms by functions of the form f(z) + g(ψ), where / and g are polynomials. The methods involve conf ormal welding and Hardy space theory. We construct a Jordan arc of positive continuous analytic capacity such that the harmonic measures for the two complementary domains are mutually absolutely continuous. Let C(S) denote the space of continuous, complex-valued functions on the unit circle S. For feC(S), let P(f) denote the space of all polynomials in / with complex coefficients. Let z denote the identity function. Browder and Wermer proved the following theorem [3, p. 551]. Let ψ be a direction-reve rsing homeomorphism of S onto S. Then the vector space sum P(z) + P(ψ) is uniformly dense in C(S). In the first part of this paper we prove a result that partially extends this theorem to the C1 norm. We say that a direction-reve rsing homeomorphism ψ:S—*S is an involution if ψoψ = z. Let C^S) denote the space of continuously-differentiable, complex-valued functions on S, with the norm

Input–Output Description of Roomy Systems

In the classical dynamic systems theory, precise information about a system can be deduced from its input–output map. In fact, for minimal systems, a complete pole-zero theory can be constructed using polynomial coprime factorization techniques, together with algebraic properties of the state space module. In this paper, an infinite-dimensional theory in the same style is presented whereby the input–output maps are assumed to exhibit some energy conservation properties and whereby these maps are ascertained to belong to a class of systems with nontrivial nullspace, called “roomy” systems. As a result, a coprime factorization theory can be deduced based not on properties of polynomials but of analytical functions, a complete polar description of the system can be given and a zero description for a somewhat more restricted class. The mathematical tools used lean heavily on Helson and Lowdenslaeger’s invariant subspace theory of Hardy spaces which quite naturally comes into play through the Bochner–Chandrase...

On bounded functions in the abstract Hardy space theory, III

On bounded functions in the abstract Hardy space theory, III

Minimum mean-square-error approximation using transversal-type filters

The classical problem of weighted minimum mean-square error approximation using transversal-type filters is considered. Results from the theory of Hardy spaces are used to establish necessary and sufficient conditions for the convergence of transversal-type filters to the optimum linear causal filter. These show that the use of second-order (and higher) all-pass approximations to a pure delay is undesirable in minimum mean-square-error approximations with a weighting function which satisfies the Paley-Wiener conditions.

On bounded functions in the abstract Hardy space theory, II

On bounded functions in the abstract Hardy space theory, II

On bounded functions in the abstract Hardy space theory

On bounded functions in the abstract Hardy space theory

On the error matrix in optimal linear filtering of stationary processes

The error covariance matrix corresponding to optimal linear causal filtering of second-order stationary processes in additive noise is considered. Formulas expressing this error matrix in terms of the optimal transfer function are established, and in the nonsingular case the optimal transfer function is expressed in terms of the spectral densities. These are straightforward generalizations of previously published scalar results, and the derivation is similarly based on Hardy space theory. Explicit bounds on the minimal error (i.e., the trace of the optimal error covariance matrix) are obtained for filtering in white noise. Furthermore, an explicit expression for the error covariance matrix is derived for the case of transmitting the same signal over several white-noise channels.

On the distribution of values of functions in some function classes in the abstract Hardy space theory

On the distribution of values of functions in some function classes in the abstract Hardy space theory

Error expressions for optimal linear filtering of stationary processes

Error expressions for optimal linear causal filtering of second-order stationary processes are derived by a method based on Hardy space theory. This method is applicable when one of the spectral densities is rational with a known structure. The operations involved in the computation of error expressions are mainly simple residue evaluations. Several examples are given.