Convolution Algebra Research Articles

The Product Formula and Convolution Structure Associated with the Generalized Hermite Polynomials

Here the product formula for the generalized and suitably normalized Hermite polynomials with parameter μ ≥ 0 will be explicitly established. Its measure turns out to be absolutely continuous and supported on two disjoint intervals lying symmetrically on the real line, provided that μ > 0. In the limit case μ = 0, which is associated with the classical Hermite polynomials, four additional point masses occur at the endpoints of the two intervals. As an application, the product formula is used to introduce a generalized translation operator and a corresponding convolution product on appropriately weighted Lebesgue spaces. To this end, norm estimates of the translation operator from above and below are presented. For any μ ≥ 1 2 , this gives rise to a quasi-positive convolution algebra.

Coideal subalgebras in hopf algebras: freeness, integrals, smash products

A. Masuoka and Y. Doi have given several necessary and sufficient conditions for a finite-dimensional Hopf algebra A to be free as a left and a right module over a right coideal subalgebra B. We derive further equivalent conditions: for example that B is a direct summand of A both as a left and a right B-module, or that B contains certain kind of elements called integrals, or that the smash products #(A,B) and #op (A,B) decompose as convolution algebras in a certain way. We study also when semisimplicity of B implies its separability.

Lattice-ordered fields as convolution algebras

Lattice-ordered fields as convolution algebras

Topological algebras withC*-enveloping algebras

LetA be a complete topological *algebra which is an inverse limit of Banach *algebras. The (unique) enveloping algebraE(A) ofA, providing a solution of the universal problem for continuous representations ofA into bounded Hilbert space operators, is known to be an inverse limit ofC*-algebras. It is shown thatS(A) is aC*-algebra iffA admits greatest continuousC*-seminorm iff the continuous states (respectively, continuous extreme states) constitute an equicontinuous set. AQ-algebra (i.e., one whose quasiregular elements form an open set)A hasC*-enveloping algebra. There exists (i) a Frechet algebra with C*-enveloping algebra that is not aQ-algebra under any topology and (ii) a non-Q spectrally bounded algebra withC*-enveloping algebra.A hermitian algebra withC*-enveloping algebra turns out to be aQ-algebra. The property of havingC*-enveloping algebra is preserved by projective tensor products and completed quotients, but not by taking closed subalgebras. Several examples of topological algebras withC*-enveloping algebras are discussed. These include several pointwise algebras of functions including well-known test function spaces of distribution theory, abstract Segal algebras and concrete convolution algebras of harmonic analysis, certain algebras of analytic functions (with Hadamard product) and Kothe sequence algebras of infinite type. The envelopingC*-algebra of a hermitian topological algebra with an orthogonal basis is isomorphic to theC*-algebrac0 of all null sequences.

Invertible elements in a convolution algebra of holomorphic functions

Invertible elements in a convolution algebra of holomorphic functions

Standard homomorphisms and convergent sequences in weighted convolution algebras

Standard homomorphisms and convergent sequences in weighted convolution algebras

Integration on loop group III. Asymptotic Peter-Weyl orthogonality

Using the Wiener measure on the continuous loop group L ( G ) of a compact connected group G, whose properties were studied in “Integration on Loop Group I,” we construct for G simply laced a stochastic central extension L ( ˜ G ) of L ( G ) . Despite the fact that this object has a multiplication defined almost everywhere, we are able to define for it a convolution algebra.

Derivations on a Fréchet convolution algebra associated with a weight

AbstractFor certain Fréchet convolution algebras associated with a weight w on the half-line [0, ∞), we are interested in the question of which Radon measures on [0, ∞) determine the derivations on the algebra. For particular weights, we show that the derivations are determined by those measures in the multiplier algebra of the original algebra. However, we also give an example of a weight for which that characterization fails. The results show a connection between the space of derivations and the behaviour for large y of the ratio w(x + y)/w(x) w(y).

Module Homomorphisms of the Dual Modules of Convolution Banach Algebras

AbstractSuppose that A is either the group algebra L1 (G) of a locally compact group G, or the Volterra algebra or a weighted convolution algebra with a regulated weight. We characterize: a) Module homomorphisms of A*, when A* is regarded an A** left Banach module with the Arens product, b) all the weak*-weak* continuous left multipliers of A**.

Harmonic analysis on solvable extensions of H-type groups

To each groupN of Heisenberg type one can associate a generalized Siegel domain, which for specialN is a symmetric space. This domain can be viewed as a solvable extensionS =NA ofN endowed with a natural left-invariant Riemannian metric. We prove that the functions onS that depend only on the distance from the identity form a commutative convolution algebra. This makesS an example of a harmonic manifold, not necessarily symmetric. In order to study this convolution algebra, we introduce the notion of “averaging projector” and of the corresponding spherical functions in a more general context. We finally determine the spherical functions for the groupsS and their Martin boundary.

Application of Characteristic Pattern Methodology to the Design of Reheat Furnace Multi-Variable Control Systems

Characteristic pattern methodology, a multi-variable control system design technique based on discrete convolution algebra, is applied to the design of a candidate computer control scheme for a steel mill reheat furnace. Time domain process input-output data are used as the basis for the process model. A robust control scheme is designed, and its performance is compared with that of a traditional decentralized three-term controller, using CBSL, a multi-variable control systems design package.

CAD of Discrete Multivariable Control Systems Using Singular Characteristic Patterns and Vectors

This paper introduces the concept of singular characteristic patterns of discrete multivariable control systems based on discrete convolution algebra. The implementation of these concepts in a new software language CBSL for simulation and computer-aided design of robust control systems is described and illustrated by the design of a robust control system for an unstable chemical reactor

Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures

Introduction. Let L be a finite dimensional Lie algebra over the real numbers, R, and let L* be its dual vector space. It is well-known [24] that the Lie algebra structure on L defines a natural Poisson structure on L*-in fact this was already known to Lie [24]-and these Poisson structures are exactly what are now called the linear Poisson structures. Given a manifold M equipped with a Poisson structure, { , one can seek deformation quantizations the direction of { , }, as first studied in [3]. These are, loosely speaking, one-parameter families, {*h}lER, of deformations of the pointwise multiplication on C<(M) (or an appropriate subalgebra), such that

Standard homomorphisms and regulated weights on weighted convolution algebras

In this paper we introduce a class of homomorphisms between weighted convolution algebras which we call standard homomorphisms and derive various equivalents of standardness. We also introduce the convergence ideal of a homomorphism. We find various descriptions of the convergence ideal together with its relation to standardness. We show that every continuous homomorphism from a weighted convolution algebra into another weighted convolution algebra, with a regulated weight, is standard, and when the algebras have both regulated weights the extension of a homomorphism to the weighted measure algebras satisfies additional continuity properties.

Amenability of discrete convolution algebras, the commutative case

Amenability of discrete convolution algebras, the commutative case

Amenability of Weighted Convolution Algebras on Locally Compact Groups

Amenability of Weighted Convolution Algebras on Locally Compact Groups

Convolutions and products of partially ordered vector-valued positive measures

The main purpose of this paper is to further investigate properties of partially ordered vector-valued measures using previous results of Wright and a positive bilinear integration process established by the author. To this end we obtain a partially ordered convolution algebra for such measures defined on the Borel σ-algebra Β(G) of a locally compact Hausdorff semigroup G. Such measures have been used by theoretical physicists to define generalized observables in quantum statistical systems and have een extensively employed in a particular formulation of Quantum Probability Theory

Automorphisms of Radical Weighted Convolution Algebras

Automorphisms of Radical Weighted Convolution Algebras

On a family of weighted convolution algebras

Continuing a line of research initiated by Larsen, Liu and Wang [12], Martin and Yap [13], Gürkanli [15], and influenced by Reiter′s presentation of Beurling and Segal algebras in Reiter [2,10] this paper presents the study of a family of Banach ideals of Beurling algebras , G a locally compact Abelian group. These spaces are defined by weighted Lp‐conditions of their Fourier transforms. In the first section invariance properties and asymptotic estimates for the translation and modulation operators are given. Using these it is possible to characterize inclusions in section 3 and to show that two spaces of this type coincide if and only if their parameters are equal. In section 4 the existence of approximate identities in these algebras is established, from which, among other consequences, the bijection between the closed ideals of these algebras and those of the corresponding Beurling algebra is derived.

Amenability of weighted convolution algebras on locally compact groups

We give a direct transition from the existence of a bounded right approximate identity in the diagonal ideal for a weighted convolution algebra on a locally compact group to the existence of translation invariant means on an associated weighted L ∞ {L^\infty } -space, thus giving a characterization of amenability for such an algebra.