Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type

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 e...

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Published in:Rocznik Akademii Górniczo-Hutniczej im. Stanisława Staszica. Opuscula Mathematica Vol. 33; no. 1; pp. 81 - 98
Main Authors: Iakovlev, Serguei I., Iakovleva, Valentina
Format: Journal Article
Language:English
Published: AGH Univeristy of Science and Technology Press 2013
Subjects:
countably normed space
eigenfunctions
eigenvalues
generator
initial function
method of steps
mixed-type differential-difference equations
spectrum
Steklov's smoothing operator
transformation group
ISSN:1232-9274
Online Access:Get full text
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Summary: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.
ISSN:1232-9274
DOI:10.7494/OpMath.2013.33.1.81