@Article{Iakovlev2013,
author = "Iakovlev, Serguei I. and Iakovleva, Valentina",
title = "Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type",
journal = "Opuscula Math.",
year = "2013",
volume = "33",
number = "1",
pages = "81--98",
doi = "http://dx.doi.org/10.7494/OpMath.2013.33.1.81",
url = "http://dx.doi.org/10.7494/OpMath.2013.33.1.81",
abstract = "It is shown that any \(\mu \in \mathbb{C}\) is an infinite multiplicity eigenvalue of the Steklov smoothing operator \(S_h\) acting on the space \(L^1_{loc}(\mathbb{R})\). For \(\mu \neq 0\) the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.",
keywords = "Steklov's smoothing operator, spectrum, eigenvalues, eigenfunctions, mixed-type differential-difference equations, initial function, method of steps, countably normed space, transformation group, generator",
issn = "1232-9274"
}