Algorithms for matrix extension and orthogonal wavelet filter banks over algebraic number fields

As a finite dimensional linear space over the rational number field Q, an algebraic number field is of particular importance and interest in mathematics and engineering. Algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics. We observe that all known finitely supported orthogonal wavelet low-pass filters in the literature have coefficients coming from an algebraic number field. Therefore, it is of theoretical and practical interest for us to consider orthogonal wavelet filter banks over algebraic number fields. In this paper, we formulate the matrix extension problem over any general subfield of C (including an algebraic number field as a special case), and we provide step-by-step algorithms to implement our main results. As an application, we obtain a satisfactory algorithm for constructing orthogonal wavelet filter banks over algebraic number fields. Several examples are provided to illustrate the algorithms proposed in this paper.