  Given a nontrivial positive measure $\mu$ on the unit circle, the associated
Christoffel-Darboux kernels are $K_n(z, w;\mu) =
\sum_{k=0}^{n}\overline{\varphi_{k}(w;\mu)}\,\varphi_{k}(z;\mu)$, $n \geq 0$,
where $\varphi_{k}(\cdot; \mu)$ are the orthonormal polynomials with respect to
the measure $\mu$. Let the positive measure $\nu$ on the unit circle be given
by $d \nu(z) = |G_{2m}(z)|\, d \mu(z)$, where $G_{2m}$ is a conjugate
reciprocal polynomial of exact degree $2m$. We establish a determinantal
formula expressing $\{K_n(z,w;\nu)\}_{n \geq 0}$ directly in terms of
$\{K_n(z,w;\mu)\}_{n \geq 0}$.
  Furthermore, we consider the special case of $w=1$; it is known that
appropriately normalized polynomials $K_n(z,1;\mu) $ satisfy a recurrence
relation whose coefficients are given in terms of two sets of real parameters $
\{c_n(\mu)\}_{n=1}^{\infty}$ and $ \{g_{n}(\mu)\}_{n=1}^{\infty}$, with
$0<g_n<1 $ for $n\geq 1$. The double sequence $\{(c_n(\mu),
g_{n}(\mu))\}_{n=1}^{\infty}$ characterizes the measure $\mu$. A natural
question about the relation between the parameters $c_n(\mu)$, $g_n(\mu)$,
associated with $\mu$, and the sequences $c_n(\nu)$, $g_n(\nu)$, corresponding
to $\nu$, is also addressed.
  Finally, examples are considered, such as the Geronimus weight (a measure
supported on an arc of the unit circle), a class of measures given by basic
hypergeometric functions, and a class of measures with hypergeometric
orthogonal polynomials.
