Near Optimal Rational Approximations of Large Data Sets

Abstra t. We introdu e a new omputationally e ient algorithm for onstru ting near optimal rational approximations of large (onedimensional) data sets. In ontrast to wavelet-type approximations, these new approximations are e e tively shift invariant. We note that the omplexity of urrent algorithms for omputing 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 rst onstru t its intermediate 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 re ently developed fast and a urate redu tion algorithm for obtaining a near optimal rational approximation from a suboptimal one. Our approa h requires rst splitting the data into large segments, whi h may later be merged together, if needed. We also des ribe a fast algorithm for evaluating these rational approximations. In parti ular, this allows us to interpolate the original data to any grid. One of the pra ti al appli ations of our algorithm is the ompression of audio signals. To demonstrate the potential ompetitiveness of our approa h, we onstru t a near optimal rational approximation of a piano re ording.