  Let X be a K\"ahler manifold, and E be a Hermitian vector bundle on X. We
investigate the space N(X,E) of nearly holomorphic sections in E, which
generalizes the notion of nearly holomorphic functions introduced by Shimura.
If X=U/K is a compact Hermitian symmetric space, and E is U-homogeneous, it
turns out that N(X,E) coincides with the space of $U$-finite vectors in
C^\infty(X,E), and we obtain new results on the U-type decomposition of the
Hilbert space of square integrable sections. As an application, we determine
this decomposition for the holomorphic tangent space of X.
