  It is known that all $k$-homogeneous orthogonally additive polynomials $P$
over $C(K)$ are of the form $$ P(x)=\int_K x^k d\mu . $$ Thus $x\mapsto x^k$
factors all orthogonally additive polynomials through some linear form $\mu$.
We show that no such linearization is possible without homogeneity. However, we
also show that every orthogonally additive holomorphic functions of bounded
type $f$ over $C(K)$ is of the form $$ f(x)=\int_K h(x) d\mu $$ for some $\mu$
and holomorphic $h\colon C(K) \to L^1(\mu)$ of bounded type.
