New Characterizations of Riesz Bases

Various equivalent conditions under which a frame is a Riesz basis of a separable Hilbert space are known. See [8] among others, for instance. We add two new conditions to the list. They are inspired by the projection method proposed in [2], which approximates frame coefficients by using finite subsets of a frame. Our main approach is to transcribe anything involving a frame in a Hilbert space into its Gram matrix acting on Euclidean spaces or on l, the Hilbert space of square summable sequences, which is isomorphic to the original space. By combining this approach with the observation in [2] that any finite subset of a Hilbert space is a frame of its linear span we acquire an equivalent condition for a finitely independent frame to be a Riesz basis in terms of the weak convergence of certain sequences which arise naturally from the frame. The weak convergence of these sequences is actually equivalent to the applicability of the projection method. Therefore it follows that the projection method works for a finitely independent frame if and only if the frame is a Riesz basis. We also observe that frame bounds of a finite linearly independent frame can be given by the eigenvalues of the Gram matrix formed by the frame. Using this observation, we have another equivalent condition under which a frame is a Riesz basis in terms of the uniform boundedness of eigenvalues of the Gram matrices of finite subsets of the frame. As a by-product we obtain formulas of Riesz bounds in terms of these eigenvalues.