Some linear models are necessarily parametric

We prove the surprising result that rather general assumptions on the set of admissible signals [xi](t) observed in the presence of additive noise [var epsilon](t) on a closed interval [a, b], imply that the set is finite dimensional, i.e., [xi](t) = [theta]1[xi]1(t) + ... + [theta]m[xi]m(t) for some integer m [greater-or-equal, slanted] 1 and fixed functions [xi]1(t),..., [xi]m(t). Thus, estimating the signal [xi](t) from observations of x(t) = [xi](t) + [var epsilon](t) reduces to estimating the parameters [theta]1,...,[theta]m. This gives a strong argument in favor of parametric linear models.