Certain Class Of Linear Operators Research Articles

On an explicit representation of the solution of linear stochastic partial differential equations with delays

Based on the analysis of a certain class of linear operators on a Banach space, we provide a closed form expression for the solutions of certain linear partial differential equations with non-autonomous input, time delays and stochastic terms, which takes the form of an infinite series expansion.

$K$-Dimensional Coding Schemes in Hilbert Spaces

This paper presents a general coding method where data in a Hilbert space are represented by finite dimensional coding vectors. The method is based on empirical risk minimization within a certain class of linear operators, which map the set of coding vectors to the Hilbert space. Two results bounding the expected reconstruction error of the method are derived, which highlight the role played by the codebook and the class of linear operators. The results are specialized to some cases of practical importance, including K-means clustering, nonnegative matrix factorization and other sparse coding methods.

A note on the Ritz method with an application to overtone stellar pulsation theory

The Ritz method reduces eigenvalue problems involving linear operators on infinite dimensional spaces to finite matrix eigenvalue problems. This paper shows that for a certain class of linear operators it is possible to choose the coordinate functions so that numerical solution of the matrix equations is considerably simplified, especially when the matrices are large. The method is applied to the problem of overtone pulsations of stars.