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A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras

100%

Zhong Huang S.

Studia Mathematica

1999

tom 132

nr 1

37-69

Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function $φ_a(t):=φ(α_t a)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum $σ_w*(φ_a)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define $Ʌ_φ^a$ to be the union of all sets $σ_w*(φ_a)$ where a ∈ A, and $Λ_α$ to be the closure of the union of all sets $Ʌ_φ^a$ where φ ∈ ∆(A), and call $Λ_α$ the unitary spectrum of α. Starting by showing that the closure of $Ʌ_φ^a$ (for fixed φ ∈ ∆(A)) is a subsemigroup of Ĝ we characterize the structure properties of the group representation α such as norm continuity, growth and existence of non-trivial invariant subspaces through its unitary spectrum $Λ_α.$ For an automorphism T of a semisimple commutative Banach algebra A we consider the group representation T: ℤ → Aut (A) defined by $T_n:=T^n$ for all n ∈ ℤ. It is shown that $Λ_T=σ(T)∩𝕋$, where σ(T) is the spectrum of T and 𝕋 is the unit circle. From this fact we give an easy proof of the Kamowitz-Scheinberg theorem which asserts that the spectrum σ(T) either contains 𝕋 or is a finite union of finite subgroups of 𝕋.

Optimal random sampling for spectrum estimation in DASP applications

100%

Tarczynski A., Qu D.

International Journal of Applied Mathematics and Computer Science

2005

tom 15

nr 4

463-469

In this paper we analyse a class of DASP (Digital Alias-free Signal Processing) methods for spectrum estimation of sampled signals. These methods consist in sampling the processed signals at randomly selected time instants. We construct estimators of Fourier transforms of the analysed signals. The estimators are unbiased inside arbitrarily wide frequency ranges, regardless of how sparsely the signal samples are collected. In order to facilitate quality assessment of the estimators, we calculate their standard deviations. The optimal sampling scheme that minimises the variance of the resulting estimator is derived. The further analysis presented in this paper shows how sampling instant jitter deteriorates the quality of spectrum estimation. A couple of numerical examples illustrate the main thesis of the paper.

Ograniczanie wyników

1 International Journal of Applied Mathematics and Computer Science

1 Studia Mathematica

1 Qu D.

1 Tarczynski A.

1 Zhong Huang S.

1 2005

1 1999

Wyniki wyszukiwania

A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras

Optimal random sampling for spectrum estimation in DASP applications