Quantified versions of Ingham's theorem

Abstract

Abstract. We obtain quantified versions of Ingham's classical Tauberian theorem and some of its variants by means of a natural modification of Ingham's own simple proof. As corollaries of the main general results, we obtain quantified decay estimates for C0-semigroups. The results reproduce those known in the literature but are both more general and, in one case, sharper. They also lead to a better understanding of the previously obscure "fudge factor" appearing in proofs based on estimating contour integrals.

Asymptotics for Infinite Systems of Differential Equations

“…The main result here is Theorem 3.1, which gives a sufficient condition for uniform boundedness involving the derivatives of φ. In Section 4, we then combine the results of Sections 2 and 3 with known results in ergodic theory and recent results in the theory of C 0 -semigroups [6,7,16] in order to obtain our main result, Theorem 4.3, which describes the asymptotic behaviour of solutions to general systems in our class. For instance, it is a consequence of Theorem 4.3 that there exists an even integer n ≥ 2 determined solely by the characteristic function φ such that for all x 0 ∈ X the derivative of the solution x(t), t ≥ 0, of (1.2) satisfies the quantified decay estimate…”

Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry

“…Recall that the Laplace of f is given by (Lf ) (λ) = R(λ, T )g for Re λ > 0. Hence, by a calculation similar to that in ( [11], Eq. (1.2)) we see that f satisfies the assumptions of Theorem 1 provided that g ∈ D(T ) ∩ Ran(T ) and R(λ, T )g extends continuously to a sufficiently smooth function on iR\ {0} .…”

“…In Section 2, we give a corollary of a quantified version of Ingham's theorem [11] which implies the rates of convergence…”

An abstract approach to optimal decay of functions and operator semigroups

“…We refer the interested reader to [15, Chapter III] for a historical overview of the Ingham-Karamata theorem and to [8,9] for recent contributions in the unquantified case. Then for any c ∈ (0, 1/2) we have Note that in the formulation of [4], which really considers the derivative of our function f , it would be natural to have an additional factor of |λ| in the numerator of (1.2), but by [6] the above weaker condition is sufficient. We remark that if M increases more rapidly than a polynomial, then the constant c in (1.3) can generally be absorbed in the O-constant.…”