  Big $q$-Jacobi functions are eigenfunctions of a second order $q$-difference
operator $L$. We study $L$ as an unbounded self-adjoint operator on an
$L^2$-space of functions on $\mathbb R$ with a discrete measure. We describe
explicitly the spectral decomposition of $L$ using an integral transform
$\mathcal F$ with two different big $q$-Jacobi functions as a kernel, and we
construct the inverse of $\mathcal F$.
