Near optimal rational approximations of large data sets

We introduce a new computationally efficient algorithm for constructing near optimal rational approximations of large (one-dimensional) data sets. In contrast to wavelet-type approximations, these new approximations are effectively shift invariant. We note that the complexity of current algorithms for computing near optimal rational approximations prevents their use for large data sets.In order to obtain a near optimal rational approximation of a large data set, we first construct its B-spline representation. Then, by using a new rational approximation of B-splines, we arrive at a suboptimal rational approximation of the data set. We then use a recently developed fast and accurate reduction algorithm for obtaining a near optimal rational approximation from a suboptimal one. Our approach requires first splitting the data into large segments, which may later be merged together, if needed. We also describe a fast algorithm for evaluating these rational approximations. In particular, this allows us to interpolate the original data to any grid.One of the practical applications of our algorithm is the compression of audio signals. To demonstrate the potential competitiveness of our approach, we construct a near optimal rational approximation of a piano recording.