  We consider a filtration of the symmetric function space given by
$\Lambda^{(k)}_t$, the linear span of Hall-Littlewood polynomials indexed by
partitions whose first part is not larger than $k$. We introduce symmetric
functions called the $k$-Schur functions, providing an analog for the Schur
functions in the subspaces $\Lambda^{(k)}_t$. We prove several properties for
the $k$-Schur functions including that they form a basis for these subspaces
that reduces to the Schur basis when $k$ is large. We also show that the
connection coefficients for the $k$-Schur function basis with the Macdonald
polynomials belonging to $\Lambda^{(k)}_t$ are polynomials in $q$ and $t$ with
integral coefficients. In fact, we conjecture that these integral coefficients
are actually positive, and give several other conjectures generalizing Schur
function theory.
