We consider the problem of constructing an optimal set of orthogonal vectors from a given set of vectors in a real Hilbert space. The vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. We show that the design of the optimal vectors, referred to as the least-squares (LS) orthogonal vectors, can be formulated as a semid...

The numerical scheme for solving the Helmholtz equation, based on the Lanczos orthogonalization scheme, is generalized so that it can be applied to media with space-dependent absorption or gain profiles.

Let μ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system {1, x, x, . . . , xn, . . . } to get orthogonal polynomials Pn(x), n ≥ 0, which have leading coefficient 1 and satisfy (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x). In general it is almost impossible to use this process to compute the explicit form of these po...

We propose a new methodology for the analysis of impulse response functions in VAR or VARMA models. More precisely, we build our results on the non ambiguous notion of innovation of a stochastic process and we consider the impact of any kind of new information at a given date t on the future values of the process. This methodology allows to take into account qualitative or quantitative informat...

Independent component analysis (ICA), instead of the traditional discrete cosine transform (DCT), is often used to project log Mel spectrum in robust speech feature extraction. The paper proposed using symmetric orthogonalization in ICA for projecting log Mel spectrum into a new feature space as a substitute in extracting speech features to solve the problem of cumulative error and unequal weig...