Basis orthonormalization procedure impacts of the basic quadratic non-uniform spline space on the scaling and wavelet functions

This paper investigates the mathematical framework of the multiresolution approach under the assumption that the sequence knots are irregularly spaced. The study is based on the construction of nested non-uniform quadratic spline multiresolution spaces. We focus on the construction of suitable quadratic orthonormal spline scaling and wavelet bases. If no more additional conditions than multiresolution ones are imposed, the orthonormal basis of the quadratic spline space is represented, on each bounded interval of the sequence, by three discontinuous scaling functions. Therefore, the quadratic spline wavelet basis, closely related to the scaling basis, is also defined by a set of discontinuous wavelet functions on each bounded interval of the sequence. We show that a judicious orthonormalization procedure of the basic quadratic spline space basis allows to (i) satisfying the continuity conditions of the scaling and wavelet functions, (ii) reducing the number of the wavelet functions to only one function, and (iii) reducing the complexity of the filter bank.