Generating Jacobi forms

In this paper we explore the relationship between vector-valued modular forms and Jacobi forms and give explicit relations over various congruence subgroups. The main result is that a Jacobi form of square-free index on the full Jacobi group is uniquely determined by any of the associated vector components. In addition, an explicit construction is given to determine the other vector components from this single component. In other words, we give an explicit construction of a Jacobi form from a subset of its Fourier coefficients. This leads to results about how the transformation properties are affected by congruence restrictions on the Fourier expansion.