This paper presents the optimum approximation theory of multiple-transmission systems expressed as matrix filter banks. With respect to any measure of error that is identical to an arbitrary operator, functional or function of elements in the corresponding error matrix of the matrix filter bank, the presented approximation is able to achieve the minimum upper limit of the measure of error (worst-case measure of error) among all the matrix filter banks using the same analysis filter matrices and the same sampler matrices. In this paper, without entering details, we assume that the ordinary inner product (a, b) between two functions a = a(x) and b = b(x) and the ordinary inner product (a, b) between two row-vectors a and b are defined already. Further, we define inner product between two matrices in the following discussion. Because we use these different types of inner products in this paper, to avoid confusion of notations, we define new notation of inner product between two matrices. For this purpose, in the first part of this paper, we borrow the well known notation of inner product < a|b > between a row-vector ( bra-vector) < a| and a column-vector (cket-vector) |b > in quantum mechanics and we extend this expression to an inner product 展开▼
