  Let A be a set of integers. For every integer n, let r_{A,h}(n) denote the
number of representations of n in the form n = a_1 + a_2 + ... + a_h, where
a_1, a_2,...,a_h are in A and a_1 \leq a_2 \leq ... \leq a_h. The function
r_{A,h}: Z \to N_0 \cup \infty is the representation function of order h for A.
The set A is called an asymptotic basis of order h if r_{A,h}^{-1}(0) is
finite, that is, if every integer with at most a finite number of exceptions
can be represented as the sum of exactly h not necessarily distinct elements of
A. It is proved that every function is a representation function, that is, if
f: Z \to N_0 \cup \infty is any function such that f^{-1}(0) is finite, then
there exists a set A of integers such that f(n) = r_{A,h}(n) for all n in Z.
Moreover, the set A can be arbitrarily sparse in the sense that, if \phi(x) \to
\infty, then there exists a set A with f(n) = r_{A,h}(n) such that card{a in A
: |a| \leq x} < \phi(x) for all sufficiently large x.
