On Restricting Cauchy-Pexider Equations to Submanifolds
Marcos Charalambides
TL;DR
This paper establishes geometric conditions under which the Cauchy-Pexider functional equation, when restricted to hypersurfaces in R^d, admits only solutions that extend uniquely to exponential affine functions, also exploring related equations.
Contribution
It provides new geometric criteria that guarantee the uniqueness and extension of solutions to the restricted Cauchy-Pexider equation on hypersurfaces.
Findings
Solutions extend uniquely to exponential affine functions under certain geometric conditions
The paper characterizes when solutions are restricted to trivial or exponential affine forms
Related functional equations are also analyzed for similar properties
Abstract
Sufficient geometric conditions are given which determine when the Cauchy-Pexider functional equation f(x)g(y)=h(x+y) restricted to x,y lying on a hypersurface in R^d has only solutions which extend uniquely to exponential affine functions. Some related functional equations are also considered.
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