Eliminating distortion in the Beylkin-Coifman-Rokhlin transform

A systematic approach is presented for the elimination of distortion in the Beylkin-Coifman-Rokhlin (BCR) transform, a technique that requires only O(N) operations to apply an N*N matrix to an arbitrary vector. Since these matrices and vectors are of finite length, implementations of the BCR transform require the application of some extension technique, and these extension methods result in an additional O(N) nonzero terms. The resulting algorithm retains O(N) complexity while eliminating all distortion in a perfect reconstruction sense. The only distortion remaining is in the wavelet coefficients, and that is due to the particular extension method chosen.<>