Boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions (Q2669355)

If \(w\in L^1(\mathbb{R}^d)\) is positive, then its Fourier transform \[ \hat w(\xi):=\int_{\mathbb{R}^d} w(x)e^{-2\pi i\; x\cdot \xi}\;dx \] determines via \(k(x,y):=\hat w(x-y)\) a unique Hilbert space \(H_k\) with reproducing kernel \(k\). Conditions are given that guaranty boundedness of composition operators on \(H_k\). Affine maps \(\phi\) play a central role. It is shown that none of the operators considered is compact. A large and very technical part of the paper is devoted to study a connection between these composition operators and asymptotic properties of the greatest zeros of orthogonal polymomials on a certain weighted \(L^2\)-space on the real line.