On close-to-scalar one-parameter cosine families

Abstract

It is shown that if two cosine families with values in a normed algebra with unity, both indexed by t running over all real numbers, of which one consists of the multiples of the unity of the algebra by numbers of the form cos at for some real a, differ in norm by less than 8/(3 √ 3) uniformly in t, then these families coincide. For a = 0, the constant 8/(3 √ 3) is optimal and cannot be replaced by any larger number.

Bounded Cosine Functions Close to Continuous Scalar Bounded Cosine Functions

“…The purpose of this paper is to show that this result holds when sup t ≥0 C (t )− c(t ) < 8 3 3 , which is the optimal constant since sup t ∈R |cos(at ) − cos(3at )| = 8 3 3 for every a = 0, and that no continuity condition on C is needed if the scalar bounded cosine function c is assumed to be continuous, see theorem 3.6 (ii). In fact, this "0 − 8 3 3 law" was already proved in a very recent paper by Bobrowski, Chojnacki and Gregoriewicz [5], which appeared after the present paper was submitted. The methods developed here vary in various aspects from those used by Bobrowski et al…”

Less than one implies zero

“…Between the first draft 1 and this version of the manuscript, Chojnacki showed in [6] that Theorem 1.1 even holds for cosine families on normed algebras indexed by general abelian groups and without assuming strong continuity. Furthermore, Bobrowski, Chojnacki and Gregosiewicz [4] and, independently, Esterle [7] extended Theorem 1.1 to r < 8 3 √ 3 ≈ 1.54. This is optimal as sup t≥0 | cos(3t) − cos(t)| = 8 3 √…”

“…In the next section we prove Theorem 1.1 for a = 0 using elementary techniques, which seem to be less involved than the technique used in [3]. As mentioned, the case a = 0 can be found in [11], see also [3,4,5,6].…”

On Wallen-type formulae for integrated semigroups and sine functions

“…Also, he established a close link with the related 0 − 3 2 law of Arendt [1]. Furthermore, in [7] and, independently but slightly later, in [14], it was proved that for κ = 0, the constant on the right-hand side of (4) may be enlarged to 8 …”