New basis orthonormalization procedure of the linear non-uniform spline scaling and wavelet spaces

This paper investigates the mathematical framework of multiresolution analysis under the assumption that the sequence knots are irregularly spaced. The study is based on the construction of nested non-uniform linear spline multiresolution spaces. We focus on the construction of suitable linear orthonormal spline scaling and wavelet bases. If no more additional conditions than multiresolution ones are imposed, the orthonormal basis of the linear spline space is represented by two discontinuous scaling functions. Therefore, the linear spline wavelet basis, closely related to the scaling basis, is defined by a set of two discontinuous wavelet functions. In addition, the orthogonal decomposition is implemented using filter banks where the coefficients depend on the location of the knots on the sequence. The main objective of this paper is to show that a judicious orthonormalization procedure of the basic linear spline space basis allows to (i) satisfying the continuity conditions of the scaling and wavelet finictions, (ii) reducing the number of the wavelet functions to only one function (iii) reducing the complexity of the filter bank.