Position-dependent Lagrange interpolating multiresolutions

This paper is devoted to the construction of interpolating multiresolutions using Lagrange polynomials and incorporating a position dependency. It uses the Harten's framework(21) and its connection to subdivision schemes. Convergence is first emphasized. Then, plugging the various ingredients into the wavelet multiresolution analysis machinery, the construction leads to position-dependent interpolating bases and multi-scale decompositions that are useful in many instances where classical translation-invariant frameworks fail. A multivariate generalization is proposed and analyzed. We investigate applications to the reduction of the so-called Gibbs phenomenon for the approximation of locally discontinuous functions and to the improvement of the compression of locally discontinuous 1D signals. Some applications to image decomposition are finally presented.