  In this series of papers we study subspaces of de Branges spaces of entire
functions which are generated by majorization on subsets $D$ of the closed
upper half-plane. The present, first, part is addressed to the question which
subspaces of a given de Branges space can be represented by means of
majorization. Results depend on the set $D$ where majorization is permitted.
Significantly different situations are encountered when $D$ is close to the
real axis or accumulates to $i\infty$.
