A recursive algorithm approximating frame coefficients related to Riesz bases of translates

In this paper, we show that frame coefficients related to Riesz bases of integer-translates are solutions of bi-infinite invertible Toeplitz systems. Hence the corresponding frame coefficients may be obtained by solving Toeplitz system directly, without complicated calculation of the inverse frame operator. We apply the finite section method to Toeplitz operator and provide a bi-directional Levinson algorithm to approximate solutions of biinfinite Toeplitz system, whose convergence is guaranteed by the recursive nature and properties of finite section method. Other advantages of the algorithm are the modest computational cost and less storage requirement.