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A study of resolvent set for a class of band operators with matrix elements

Andrey Osipov (2016)

Concrete Operators

For operators generated by a certain class of infinite three-diagonal matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying second order finite-difference equations. This enables us to describe some asymptotic behavior of the corresponding systems of vector orthogonal polynomials on the resolvent set. We also find that the operators generated by infinite Jacobi matrices have the largest resolvent set in this class.

An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

Antoine Gloria, Stefan Neukamm, Felix Otto (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm...

Characterization of functions whose forward differences are exponential polynomials

J. M. Almira (2017)

Commentationes Mathematicae Universitatis Carolinae

Given { h 1 , , h t } a finite subset of d , we study the continuous complex valued functions and the Schwartz complex valued distributions f defined on d with the property that the forward differences Δ h k m k f are (in distributional sense) continuous exponential polynomials for some natural numbers m 1 , , m t .

Classification rationnelle et confluence des systèmes aux différences singuliers réguliers

Julien Roques (2006)

Annales de l’institut Fourier

En choisissant des “caractères” et des “logarithmes”, méromorphes sur , construits à l’aide de la fonction Gamma d’Euler, et en utilisant des séries de factorielles convergentes, nous sommes en mesure, dans une première partie, de donner une “forme normale” pour les solutions d’un système aux différences singulier régulier. Nous pouvons alors définir une matrice de connexion d’un tel système. Nous étudions ensuite, suivant une idée de G.D. Birkhoff, le lien de celles-ci avec le problème de la classification...

Criterion of p -criticality for one term 2 n -order difference operators

Petr Hasil (2011)

Archivum Mathematicum

We investigate the criticality of the one term 2 n -order difference operators l ( y ) k = Δ n ( r k Δ n y k ) . We explicitly determine the recessive and the dominant system of solutions of the equation l ( y ) k = 0 . Using their structure we prove a criticality criterion.

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