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On the asymptotic convergence of the polynomial collocation method for singular integral equations and periodic pseudodifferential equations

A. I. Fedotov (2002)

Archivum Mathematicum

We prove the convergence of polynomial collocation method for periodic singular integral, pseudodifferential and the systems of pseudodifferential equations in Sobolev spaces H s via the equivalence between the collocation and modified Galerkin methods. The boundness of the Lagrange interpolation operator in this spaces when s > 1 / 2 allows to obtain the optimal error estimate for the approximate solution i.e. it has the same rate as the best approximation of the exact solution by the polynomials.

On the Cauchy problem in linear viscoelasticity

Pasquale Renno (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Con riferimento all’operatore integrodifferenziale della viscoelasticità lineare nella formulazione creep, si determina la soluzione fondamentale E in corrispondenza di un’arbitraria funzione di memoria. Di conseguenza viene risolto esplicitamente il problema di Cauchy relativo al moto unidimensionale di un sistema viscoelastico ℬ , omogeneo ed isotropo, determinato da dati iniziali e storia di stress comunque prefissati. Successivamente, nell’ambito di opportune ipotesi di memoria labile, si dimostrano...

On the eigenvalues of a class of hypo-elliptic operators. IV

Johannes Sjöstrand (1980)

Annales de l'institut Fourier

Let P be a selfadjoint classical pseudo-differential operator of order > 1 with non-negative principal symbol on a compact manifold. We assume that P is hypoelliptic with loss of one derivative and semibounded from below. Then exp ( - t P ) , t ≥ 0 , is constructed as a non-classical Fourier integral operator and the main contribution to the asymptotic distribution of eigenvalues of P is computed. This paper is a continuation of a series of joint works with A. Menikoff.

On the Fefferman-Phong inequality

Abdesslam Boulkhemair (2008)

Annales de l’institut Fourier

We show that the number of derivatives of a non negative 2-order symbol needed to establish the classical Fefferman-Phong inequality is bounded by n 2 + 4 + ϵ improving thus the bound 2 n + 4 + ϵ obtained recently by N. Lerner and Y. Morimoto. In the case of symbols of type S 0 , 0 0 , we show that this number is bounded by n + 4 + ϵ ; more precisely, for a non negative symbol a , the Fefferman-Phong inequality holds if ∂ x α ∂ ξ β a ( x , ξ ) are bounded for, roughly, 4 ≤ | α | + | β | ≤ n + 4 + ϵ . To obtain such results and others, we first prove an abstract result which says that...

On the Hardy-type integral operators in Banach function spaces.

Elena Lomakina, Vladimir Stepanov (1998)

Publicacions Matemàtiques

Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.

On the power boundedness of certain Volterra operator pencils

Dashdondog Tsedenbayar (2003)

Studia Mathematica

Let V be the classical Volterra operator on L²(0,1), and let z be a complex number. We prove that I-zV is power bounded if and only if Re z ≥ 0 and Im z = 0, while I-zV² is power bounded if and only if z = 0. The first result yields | | ( I - V ) ⁿ - ( I - V ) n + 1 | | = O ( n - 1 / 2 ) as n → ∞, an improvement of [Py]. We also study some other related operator pencils.

On the singular numbers for some integral operators.

Alexander Meskhi (2001)

Revista Matemática Complutense

Two-sided estimates of Schatten-von Neumann norms for weighted Volterra integral operators are established. Analogous problems for some potential-type operators defined on Rn are solved.

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