  In this paper, we specify what functions induce the bounded composition
operators on a reproducing kernel Hilbert space (RKHS) associated with an
analytic positive definite function defined on $\mathbf{R}^d$. We prove that
only affine transforms can do so in a pretty large class of RKHS. Our result
covers not only the Paley-Wiener space on the real line, studied in previous
works, but also much more general RKHSs corresponding to analytic positive
definite functions where existing methods do not work. Our method only relies
on an intrinsic properties of the RKHSs, and we establish a connection between
the behavior of composition operators and the asymptotic properties of the
greatest zeros of orthogonal polynomials on a weighted $L^2$-spaces on the real
line. We also investigate the compactness of the composition operators and show
that any bounded composition operators cannot be compact in our situation.
