  We introduce here a generalization of the modified Bernstein polynomials for
Jacobi weights using the $q$-Bernstein basis proposed by G.M. Phillips to
generalize classical Bernstein Polynomials. The function is evaluated at points
which are in geometric progression in $]0,1[$. Numerous properties of the
modified Bernstein Polynomials are extended to their $q$-analogues:
simultaneous approximation, pointwise convergence even for unbounded functions,
shape-preserving property, Voronovskaya theorem, self-adjointness. Some
properties of the eigenvectors, which are $q$-extensions of Jacobi polynomials,
are given.
