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

Composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions