𝔖 Bobbio Scriptorium
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Mimetic methods on measure chains

✍ Scribed by M. Bohner; J.E. Castillo

Book ID
104352198
Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
273 KB
Volume
42
Category
Article
ISSN
0898-1221

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✦ Synopsis

We introduce the divergence and the gradient for functions defined on a measure chain, and this includes as special cases both continuous derivatives and discrete forward differences. It is shown that in one dimension, subject to Dirichlet boundary conditions, the divergence and the gradient are negative adjoints of each other and that the divergence of the gradient is negative semidefinite. These are well-known results in tile continuous theory, and hence, mimic those properties also for the case of a general measure chain.

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In this paper we show how the recently introduced concept of regular variation on time scales (or measure chains) is related to a Karamata type definition. We also present characterization theorems and an embedding theorem for regularly varying functions defined on suitable subsets of reals. We demo