Newman ’ s proof of Wiener ’ s Theorem by E B Davies 21 March 2005

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 ≤ c g n 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 ≤ c g n 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 = ∞ n=0 (g − f) n g −n−1 .