Affiliation:
1. Departamento de Ingeniería y Tecnología de Computadores, Universidad de Murcia, Murcia, Spain
Abstract
We consider Aichinger?s equation f (x1 +... + xm+1) = Xm+1 i=1 1i(x1, x2,
...,bxi,..., xm+1) for functions defined on commutative semigroups
which take values on commutative groups. The solutions of this equation are,
under very mild hypotheses, generalized polynomials. We use the canonical
form of generalized polynomials to prove that compositions and products of
generalized polynomials are again generalized polynomials and that the
bounds for the degrees are, in this new context, the natural ones. In some
cases, we also show that a polynomial function defined on a semigroup can
uniquely be extended to a polynomial function defined on a larger group. For
example, if f solves Aichinger?s equation under the additional restriction
that x1,..., xm+1 ? Rp +, then there exists a unique polynomial function
F defined on Rp such that F|R p + = f . In particular, if f is also bounded
on a set A ? Rp + with positive Lebesgue measure then its unique polynomial
extension F is an ordinary polynomial in p variables with total degree ? m,
and the functions 1i are also restrictions to Rpm + of ordinary polynomials
of total degree ? m defined on Rpm.
Publisher
National Library of Serbia