Boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions

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 $\mathbf{R}^d$. We prove that only affine transforms can do so in pretty large class of RKHS. Our result covers not Paley-Wiener real line, studied previous works, but also much more general RKHSs corresponding to where existing methods work. method relies intrinsic properties RKHSs, and establish connection between behavior asymptotic greatest zeros orthogonal polynomials weighted $L^2$-spaces line. investigate compactness show any cannot be compact our situation.