Square summability with geometric weight for classical orthogonal expansions

Let $f_k$ be the $k$-th Fourier coefficient of a function $f$ in terms of the orthonormal Hermite, Laguerre or Jacobi polynomials. We give necessary and sufficient conditions on $f$ for the inequality ${\sum }_k\mathrm{\mid }{f}_k{\mathrm{\mid }}^2{\theta }^k<\mathrm{\infty }$ to hold with $\theta >1$. As a by-product new orthogonality relations for the Hermite and Laguerre polynomials are found. The basic machinery for the proofs is provided by the theory of reproducing kernel Hilbert spaces.