$C_0$-semigroups and resolvent operators approximated by Laguerre expansions

Abstract

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 C0semigroups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions, its Laplace transforms and the convergence of Laguerre series in Lebesgue spaces. The concluding section of this paper is devote to consider some examples of C0-semigroups: shift, convolution and holomorphic semigroups where some of these results are improved.

Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive Convolution

“…Convergence. This expansion has been considered in the context of C 0 − semigroups [1,21], where − ∂xx α 2 is replaced with a general closed operator A on a Hilbert space X. In our notation, we restate part (ii) of Theorem 4.3 in [1], which is proven therein.…”

“…This strategy has recently been developed in [6] for the wave equation by the present authors. In the present work, we not only extend the method of lines transpose to parabolic problems, but we recognize the resulting expansion as a so-called resolvent expansion [1,21], which we leverage to prove stability and convergence of the successive convolution series. In addition, we incorporate nonlinear terms with an integrating factor method that relies on high order Hermite-Birkhoff interpolants as well as the (linear) resolvent expansions developed in this paper.…”

Hermite expansions of C0-groups and cosine functions

“…143]. Moreover, Hermite expansions of Dirac distribution and the distribution principal value of 1 x may be found in [2, pp 191-193] and [11,Section 2]. Hermite expansions of products of temperated distributions are considered in detail in [11].…”

“…In the literature, there exist many different approximations of C 0 -semigroups, as Euler, Yosida, Dunford-Segal or subdiagonal Padé approximations, see [15] and references in [1]. However there are not some many approximations of C 0 -groups and cosine functions: stable rational approximations for exponential function are considered to treat hyperbolic problems, i.e., C 0groups in [9] and cosine functions in [19,Section 4].…”

Quadrature Rules for L 1-Weighted Norms of Orthogonal Polynomials

“…These functions q n,ω are fundamental in classical orthogonal expansions ([11,Chapter 4] and [14, Chapter IX]). Recently the authors have treated them to introduce Laguerre expansions for C 0 -semigroups in [1] and Hermite expansions for C 0 -groups and cosine function in [2]. In fact, to get sharp estimations of q n,ω 1 is the motivating starting-point of this paper: sharp estimations allow to assure convergence of vector-valued orthogonal expansions, see more details in [1,2].…”