2014/07/15 by Adam J. Ostaszewski, Ostaszewski, Adam J.
Mathematics · #33B99 #34D05 #39A20 #39B22 #39B62 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 26A03 #math.CA #msc:26A03 #msc:33B99 #msc:34D05 #msc:39A20 #msc:39B22 #msc:39B62
paper · pdf · doi:10.48550/arxiv.1407.4089
Sequel to:N. H. Bingham and A. J. Ostaszewski, Cauchy's functional equation and extensions: Goldie's equation and inequality, the Gołąb-Schinzel equation and Beurling's equation, arxiv.org/abs/1405.3947 Related to: N. H. Bingham and A. J. Ostaszewski, Beurling moving averages and approximate homomorphisms. Redrafted with additional results
arxiv created 2014/11/07 · arxiv updated 2014/11/10
The theory of regular variation, in its Karamata and Bojanić-Karamata/de Haan forms, is long established and makes essential use of the Cauchy functional equation. Both forms are subsumed within the recent theory of Beurling regular variation, developed elsewhere. Various generalizations of the Cauchy equation, including the Gołąb-Schinzel functional equation (GS), are prominent there. Here we unify their treatment by `algebraicization': extensive use of group structures introduced by Popa and Javor in the 1960s turn all the various solutions into homomorphisms, and show that (GS) is present everywhere, even if in a thick disguise.