Yes, because the intersection of the resolution spaces is taken in L²; but polynomials do not have a finite energy. Hence, trying to analyse a polynomial of degree n with (orthogonal, to symplify) wavelets with p>=n vanishing moments:

yields a zero result. On the other hand, the decomposition

does reconstruct the polynomial, with zero coefficient on the second line of the decomposition.

In fact, the second formula is more general and can represent functions which do not belong to L².