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Author Davies, E. B.
Source CiteSeerX
Content type Text
File Format PDF
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract Since Wiener’s original proof of his theorem about periodic functions whose Fourier series are absolutely summable, a variety of quite different proofs have been devised. Some of these are described in [2, Sect. B.9.4]. In this paper we rewrite the beautiful proof of Newman [1] in a slightly more general form, so that it may be readily applied to a variety of related problems. Although we use the language of commutative Banach algebras, we do not use Gelfand’s representation theory, but construct the inverse directly using completely elementary methods. Theorem 1 Let X be a compact Hausdorff space and let A be a subalgebra of C(X) that contains the constants. Suppose that A is a Banach algebra with respect to a norm ‖ · ‖, and that D is a dense subset of A. Let k, c be positive constants and suppose that for every g ∈ D satisfying g(x) ≥ σ for some σ> 0 and all x ∈ X, g is invertible in A and ‖g −n ‖ ≤ cgn k c n σ −n (1) for all positive integers n. Then every f ∈ A which is invertible in C(X) is also invertible in A and the norm of its inverse is effectively computable. Proof If f ∈ A and z > ‖f ‖ then (z − f) is invertible in A and therefore also invertible in C(X). This implies that ‖f‖ ∞ ≤ ‖f ‖ for all f ∈ A. If f ∈ A and f(x) ≥ σ> 0 for all x ∈ X, let g ∈ D satisfy ‖g − f ‖ < δσ; we put δ = {2(1 + c)} −1. This implies that g(x) ≥ (1 − δ)σ> 0 for all x ∈ X. Therefore g is invertible in A and ‖g −n ‖ ≤ cgn k c n (1 − δ) −n σ −n for all positive integers n. The inverse of f in A is given by the formula f −1 ∞∑ = (g − f) n=0 n g −n−1. 1 This is norm convergent in A, and hence also uniformly convergent in C(X), with ‖f −1 ‖
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article
Publisher Date 2005-01-01