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We develop a general, functional calculus approach to approximation of $C_0$-semigroups on Banach spaces by bounded completely monotone functions of their generators. The approach comprises most of well-known approximation formulas, yields…
In this paper we introduce Laguerre expansions to approximate vector-valued functions expanding on the well-known scalar theorem. We apply this result to approximate $C_0$-semi\-groups and resolvent operators in abstract Banach spaces. We…
By functional calculus methods, we obtain optimal convergence rates in Euler's approximation formula for C_0-semigroups restricted to ranges of generalized Stieltjes functions. Our results include a number of partial cases studied in the…
We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known…
Using a recently developed $\mathcal H$-calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on…
We study functional calculus properties of $C_{0}$-groups on real interpolation spaces, using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference…
We improve the classical results by Brenner and Thom\'ee on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, provide sharper stability estimates, and…
Let $(r_{n})_{n \in \mathbb{N}}$ be the sequence of subdiagonal Pad\'{e} approximations of the exponential function. We prove that for $-A$ the generator of a uniformly bounded $C_{0}$-semigroup $T$ on a Banach space $X$, the sequence…
We study best approximations in Banach spaces via Birkhoff-James orthogonality of functionals. To exhibit the usefulness of Birkhoff-James orthogonality techniques in the study of best approximation problems, some algorithms and distance…
This paper studies approximation properties of linear sampling operators in general Banach lattices $X$. We obtain matching direct and inverse approximation estimates, convergence criteria, equivalence results involving special…
We study best approximations to compact operators between Banach spaces and Hilbert spaces, from the point of view of Birkhoff-James orthogonality and semi-inner-products. As an application of the present study, some distance formulae are…
We consider a Trotter-type-product formula for approximating the solution of a linear abstract Cauchy problem (given by a strongly continuous semigroup), where the underlying Banach space is a product of two spaces. In contrast to the…
The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. We present quantitative estimates for the rate of the convergence of the finite section method on weighted $\ell…
This article investigates the convergence properties of s-numbers of certain truncations of bounded linear operators between Banach spaces. We prove a generalized version of a known convergence result for the approximation numbers of…
Arbitrary operator A on a Banach space X which is the generator of C_0-group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse…
The theory of Banach spaces of Dirichlet series has drawn an increasing attention in the recent 25 years. One of the main interest of this new theory is that of defining analogues of the classical spaces of analytic functions on the unit…
We study linear operators $T$ on Banach spaces for which there exists a $C_0$-semigroup $(T(t))_{t\geq 0}$ such that $T=T(1)$. We present a necessary condition in terms of the spectral value 0 and give classes of examples where this can or…
The purpose of this work is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such…
In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We…
We study the functional calculus properties of generators of $C_{0}$-groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let $-iA$ generate a $C_{0}$-group on a Banach space $X$…