  We prove a number of quadratic transformations of elliptic Selberg integrals
(conjectured in an earlier paper of the author), as well as studying in depth
the "interpolation kernel", an analytic continuation of the author's elliptic
interpolation functions which plays a major role in the proof as well as acting
as the kernel for a Fourier transform on certain elliptic double affine Hecke
algebras (discussed in a later paper). In the process, we give a number of
examples of a new approach to proving elliptic hypergeometric integral
identities, by reduction to a Zariski dense subset of a formal neighborhood of
the trigonometric limit.
