arXiv subjects

Pedro J. Miana

Publications and source records attributed to Pedro J. Miana.

25 records · Page 2Linked to original sources

arXiv2014

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

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 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 $C_0$-semigroups: shift, convolution and holomorphic semigroups where some of these results are improved.

math.FA↗

arXiv2014

Hermite expansions of $C_0$-groups and cosine functions

In this paper we introduce vector-valued Hermite expansions to approximate one-parameter operator families such as $C_0$-groups and cosine functions. In both cases we estimate the rate of convergence of these Hermite expansions to the related family and compare with other known approximations. Finally we illustrate our results with particular examples of $C_0$-groups and cosine functions and their Hermite expansions.

math.FA↗

arXiv2014

Quadrature rules for $L^1$-weighted norms of orthogonal polynomials

In this paper we obtain $L^1$-weighted norms of classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of these orthogonal polynomials; these expressions are usually known as quadrature rules. In particular these new formulae are useful to calculate directly some positive defined integrals as several examples show.

math.CA↗

arXiv2013

On the boundedness of generalized Cesàro operators on Sobolev spaces

For $β>0$ and $p\ge 1$, the generalized Cesàro operator $$ \mathcal{C}_βf(t):=\fracβ{t^β}\int_0^t (t-s)^{β-1}f(s)ds $$ and its companion operator $\mathcal{C}_β^*$ defined on Sobolev spaces $\mathcal{T}_p^{(α)}(t^α)$ and $\mathcal{T}_p^{(α)}(| t|^α)$ (where $α\ge 0$ is the fractional order of derivation and are embedded in $L^p(\RR^+)$ and $L^p(\RR)$ respectively) are studied. We prove that if $p>1$, then $\mathcal{C}_β$ and $\mathcal{C}_β^*$ are bounded operators and commute on $\mathcal{T}_p^{(α)}(t^α)$ and $\mathcal{T}_p^{(α)}(| t|^α)$. We show explicitly the spectra $σ(\mathcal{C}_β)$ and $σ(\mathcal{C}_β^*)$ and its operator norms (which depend on $p$). For $1< p\le 2$, we prove that $ \hat{{\mathcal C}_β(f)}={\mathcal C}_β^*(\hat{f})$ and $\hat{{\mathcal C}_β^*(f)}={\mathcal C}_β(\hat{f})$ where $\hat{f}$ is the Fourier transform of a function $f\in L^p(\RR)$.

math.FA↗

arXiv2013

Sharp extensions for convoluted solutions of wave equations

In this paper we give sharp extensions of convoluted solutions of wave equations in abstract Banach spaces. The main technique is to use the algebraic structure, for convolution products $\ast$ and $\ast_c$, of these solutions which are defined by a version of the Duhamel's formula. We define algebra homomorphisms, for the convolution product $\ast_c$, from a certain set of test-functions and apply our results to concrete examples of abstract wave equations.

math.FA↗

arXiv2013

Sharp extensions for convoluted solutions of abstract Cauchy problems

In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of algebraic structure (for usual convolution product $\ast$) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of $k$-distribution semigroups to extend previous concepts of distribution semigroups.

math.FA↗

arXiv2009

$H^\infty$-functional calculus and models of Nagy-Foiaş type for sectorial operators

We prove that a sectorial operator admits an H-infty - functional calculus if and only if it has a functional model of Nagy-Foias type. Furthermore, we give a concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any sectorial operator by passing to a different norm (the McIntosh square function norm). We also show that this quadratic norm is close to the original one, in the sense that there is only a logarithmic gap between them.

math.SP↗