Piecewise lexsegment ideals

The problem of describing the Hilbert functions of homogeneous ideals of a commutative polynomial ring containing a fixed monomial ideal $I$ is considered. For this purpose the notion of a piecewise lexsegment ideal is introduced generalizing the notion of a lexsegment ideal. It is proved that if $I$ is a piecewise lexsegment ideal, then it is possible to describe the Hilbert functions of homogeneous ideals containing $I$ in a way similar to that suggested by Macaulay for the situation $I=0$. Moreover, a generalization of extremal properties of lexsegment ideals is obtained (the inequality for the Betti numbers, behaviour under factorization by homogeneous generic forms).