Tchebycheff systems and best partial bases
Oved Shisha · Pacific Journal of Mathematics · 1980
This is a contribution to the partial basis problem and, in particular, to the case where the basis elements are cosigns or consecutive powers cosines.We contribute also to the general theory of Tchebycheff systems to which the partial basis problem is strongly related.l Introduction* The partial basis problem was formulated and studied by J. T. Lewis, D. W. Tufts and the author in 1975 in connection with their study of optimization of multichannel processing.Let X be a normed linear space, let f,h u h 2 , ---,h N eX and let n be an integer, 1 <; n < N.For every sequence μ -{μ k }? of integers, with 1 <;where the minimum is taken over all possible choices of the scalars c u --,c n .The problem is to minimize e{μ).It is of particular interest when X is one of the standard function spaces.Subsequently, progress has been made both in theory and in the computational aspect.An algorithm, numerical examples and some theoretical results have been given by K. M. Levasseur and J. T. Lewis in [6].G. G. Lorentz [5] has observed that, for X-L 2