David Seifert

Rates of decay in the classical Katznelson-Tzafriri theorem

The Katznelson-Tzafriri Theorem states that, given a power-bounded operator T, ǁTn(I − T)ǁ → 0 as n → ∞ if and only if the spectrum σ(T) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T) ∩ T = {1}. The results obtained lead, in particular, to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(eiθ, T) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.

Optimal energy decay in a one-dimensional coupled wave–heat system

We study a simple one-dimensional coupled wave–heat system and obtain a sharp estimate for the rate of energy decay of classical solutions. Our approach is based on the asymptotic theory of C0-semigroups and in particular on a result due to Borichev and Tomilov (Math Ann 347:455–478, 2010), which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the semigroup generator. This technique not only leads to an optimal result, it is also simpler than the methods used by other authors in similar situations.

Asymptotics for Infinite Systems of Differential Equations

This paper investigates the asymptotic behavior of solutions to certain infinite systems of ordinary differential equations. In particular, we use results from ergodic theory and the asymptotic theory of

A quantified Tauberian theorem for sequences

C_0

Ritt operators and convergence in the method of alternating projections

-semigroups to obtain a characterization, in terms of convergence of certain Cesaro averages, of those initial values which lead to convergent solutions. Moreover, we obtain estimates on the rate of convergence for solutions whose initial values satisfy a stronger ergodic condition. These results rely on a detailed spectral analysis of the operator describing the system, which is made possible by certain structural assumptions on the operator. The resulting class of systems is sufficiently broad to cover a number of important applications including, in particular, both the so-called robot rendezvous problem and an important class of platoon systems arising in control theory. Our method leads to new results in both cases.

Some improvements of the Katznelson- Tzafriri theorem on Hilbert space

The main result of this paper is a quantified version of Inghams Tauberian theorem for bounded vector-valued sequences rather than functions. It gives an estimate on the rate of decay of such a sequence in terms of the behaviour of a certain boundary function, with the quality of the estimate depending on the degree of smoothness this boundary function is assumed to possess. The result is then used to give a new proof of the quantified Katznelson-Tzafriri theorem recently obtained in [21].

Asymptotic Behaviour of Coupled Systems in Discrete and Continuous Time

Given N ? 2 closed subspaces M 1 , ? , M N of a Hilbert space X , let P k denote the orthogonal projection onto M k , 1 ? k ? N . It is known that the sequence ( x n ) , defined recursively by x 0 = x and x n + 1 = P N ? P 1 x n for n ? 0 , converges in norm to P M x as n ? ∞ for all x ? X , where P M denotes the orthogonal projection onto M = M 1 ? ? ? M N . Moreover, the rate of convergence is either exponentially fast for all x ? X or as slow as one likes for appropriately chosen initial vectors x ? X . We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number α 0 , a dense subset X α of X such that ? x n - P M x ? = o ( n - α ) as n ? ∞ for all x ? X α . Furthermore, there exists another dense subset X ∞ of X such that, if x ? X ∞ , then ? x n - P M x ? = o ( n - α ) as n ? ∞ for all α 0 . These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that P M x is in fact the limit of a series which converges unconditionally.

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

This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations.

Non-optimality of the Greedy Algorithm for Subspace Orderings in the Method of Alternating Projections

This paper investigates the asymptotic behaviour of solutions to certain infinite systems of coupled recurrence relations. In particular, we obtain a characterisation of those initial values which lead to a convergent solution, and for initial values satisfying a slightly stronger condition we obtain an optimal estimate on the rate of convergence. By establishing a connection with a related problem in continuous time, we are able to use this optimal estimate to improve the rate of convergence in the continuous setting obtained by the authors in a previous paper. We illustrate the power of the general approach by using it to study several concrete examples, both in continuous and in discrete time.

Asymptotic behaviour in the robot rendezvous problem

Abstract This work deals with free transport equations with partly diffuse stochastic boundary operators in slab geometry. Such equations are governed by stochastic semigroups in L 1 spaces. We prove convergence to equilibrium at the rate O ( t − k 2 ( k + 1 ) + 1 ) ( t → + ∞ ) for L 1 initial data g in a suitable subspace of the domain of the generator T where k ∈ N depends on the properties of the boundary operators near the tangential velocities to the slab. This result is derived from a quantified version of Inghams tauberian theorem by showing that F g ( s ) : = lim e → 0 + ⁡ ( i s + e − T ) − 1 g exists as a C k function on R \ { 0 } such that ‖ d j d s j F g ( s ) ‖ ≤ C | s | 2 ( j + 1 ) near s = 0 and bounded as | s | → ∞ ( 0 ≤ j ≤ k ) . Various preliminary results of independent interest are given and some related open problems are pointed out.