Ultrafunctions, Projective Function Geometry, and Polynomial Functional Equations

Some non‐linear functional equations involving iterates have fascinating properties. In this paper, the polynomial functional equations (P)f N ( x ) = ∑ n = 0 N ‐ 1A f n ( x )are studied, where the A n are real constants, and the powers denote the iterates of f : R → R. Each such equation can be interpreted as a linear invariance condition for a graph Γ ⊂ R × R N −1 associated with f . The attempt to express this condition in simple canonical form leads to the idea that functions (or, rather, graphs) can be interpreted as projective objects. This motivates the introduction of the space of ultrafunctions , a ‘projective function space’ obtained by adjoining suitable ‘functions at infinity’ to the affine space of ordinary functions. This enables (P) to be reduced, by a projective transformation, to a purely linear ultrafunctional equation which is easily solved. Explicit formulae, involving ultrafunctions from an explicitly described family, are given for the general continuous (ordinary function) solution in representative generic cases. The relationship between these solutions and known ‘explicit functions’ is discussed. Further applications are briefly described; in particular, ‘functional continued fractions’ are introduced, and their convergence is examined.