133 research outputs found
Remarks on the Cauchy functional equation and variations of it
This paper examines various aspects related to the Cauchy functional equation
, a fundamental equation in the theory of functional
equations. In particular, it considers its solvability and its stability
relative to subsets of multi-dimensional Euclidean spaces and tori. Several new
types of regularity conditions are introduced, such as a one in which a complex
exponent of the unknown function is locally measurable. An initial value
approach to analyzing this equation is considered too and it yields a few
by-products, such as the existence of a non-constant real function having an
uncountable set of periods which are linearly independent over the rationals.
The analysis is extended to related equations such as the Jensen equation, the
multiplicative Cauchy equation, and the Pexider equation. The paper also
includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the
acknowledgments section in the official paper exists, but it appears before
the appendix and not before the references as in the arXiv version);
correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of
Theorem 2.1; a few small improvements in various sections; added thank
Spectra and fine spectra of certain lower triangular double-band matrices as operators on c 0
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