APPROXIMATE IDENTITY OF CONVOLUTION BANACH ALGEBRAS

Abstract

A weight ω on the positive half real line [0, ∞) is a positive continuous function such that ω(s + t) ≤ ω(s)ω(t), for all s, t ∈ [0, ∞), and ω(0) = 1. The weighted convolution Banach algebra L1(ω) is the algebra of all equivalence classes of Lebesgue measurable functions f such that (cid:107)f (cid:107) = ∞ |f (t)| ω(t)dt < ∞, under pointwise addition, scalar multiplication of functions, and the convolution product (f ∗ g)(t) = t 0 f (t − s)g(s)ds. We give a sufficient condition on a weight function ω(t) in order that L1(ω) has a bounded approximate identity.

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