Addition theorems and binary expansions

Title: Addition theorems and binary expansions
Creator: Borwein, Jonathan M.; Girgensohn, Roland
Relation: Canadian Journal of Mathematics Vol. 47, Issue 2, p. 262-273
Publisher Link: http://dx.doi.org/10.4153/CJM-1995-013-4
Publisher: University of Toronto Press
Resource Type: journal article
Date: 1995
Description: Let an interval / ⊂ ℝ and subsets D₀,D₁ ⊂ I with D₀⋃D₁ = I and D₀ ∩ D₁ = 0 be given, as well as functions r₀:D₀ → I, r₁:D₁ → I. We investigate the system (S) of two functional equations for an unknown function ∫: I → [0, 1]: [formula cannot be replicated] We derive conditions for the existence, continuity and monotonicity of a solution. It turns out that the binary expansion of a solution can be computed in a simple recursive way. This recursion is algebraic for, e.g., inverse trigonometric functions, but also for the elliptic integral of the first kind. Moreover, we use (S) to construct two kinds of peculiar functions: surjective functions whose intervals of constancy are residual in I, and strictly increasing functions whose derivative is 0 almost everywhere.
Subject: binary expansions; functional equations; addition theorems; recursions
Identifier: http://hdl.handle.net/1959.13/940917
Identifier: uon:13127
Identifier: ISSN:0008-414X
Language: eng
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