A Note on Orthogonality of The Finite Element Multiwavelets

Finite element multiwavelets were constructed by Strela and Strang [12]. They simultaneously achieve higher regularity and short support. However, these multiwavelets are not orthogonal with respect to translates. In this notes, we introduced an alternate Sobolev type inner product in which the scaling functions are orthogonal to their translates. We then derive a simple version of the decomposition and reconstruction algorithm which can be used to reduce the amount of calculation in the approximation. By the end, these new algorithms are illustrated by examples.