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Unifying order structures for Colombeau algebras
Author(s) -
Giordano Paolo,
Nigsch Eduard A.
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400277
Subject(s) - mathematics , pointwise , generalized function , pure mathematics , formalism (music) , algebra over a field , set (abstract data type) , scope (computer science) , mathematical analysis , computer science , art , musical , visual arts , programming language
We define a general notion of set of indices which, using concepts from pre‐ordered sets theory, permits to unify the presentation of several Colombeau‐type algebras of nonlinear generalized functions. In every set of indices it is possible to generalize Landau's notion of big‐O such that its usual properties continue to hold. Using this generalized notion of big‐O, these algebras can be formally defined the same way as the special Colombeau algebra. Finally, we examine the scope of this formalism and show its effectiveness by applying it to the proof of the pointwise characterization in Colombeau algebras.