Nonnegative scaling vectors on the interval (Q402323)

A method is outlined to construct nonnegative scaling vector functions for \(L^2(I)\), \(I=[a,b]\) starting from a scaling vector for a multiresolution analysis of \(L^2(\mathbb{R})\). The goal is to produce a nonnegative scaling vector for \(L^2(I)\) with the same polynomial accuracy as the original scaling vector, using the smallest possible number of edge functions. Let \(\Phi=(\phi^1,\dots ,\phi^{A})\) be a scaling vector that reproduces polynomials up to degree \(m\) and let \(f_{n,k}^\ell\) be coefficients in the expansion NEWLINE\[NEWLINEt^m=\sum_{\ell=1}^A\sum_{k\in\mathbb{Z}} f_{n,k}^\ell\phi^{\ell}(t-k) =\sum_{k\in\mathbb{Z}} {\vec{f}}_{n,k}\cdot \Phi(\cdot-k)\;. NEWLINE\]NEWLINE One defines left edge functions NEWLINE\[NEWLINE\phi_{L,n}(t)=\sum_{\ell=1}^A\sum_{k=1-M_\ell}^0 f_{n,k}^\ell\overline{\phi}^{\ell}(t-k) NEWLINE\]NEWLINE with \(M_\ell\) chosen so that \(\phi^\ell\) is supported in \([0, M_\ell]\) so the sum is restricted to those shifts of \(\phi^\ell\) supported in \([0,\infty)\), \(n=0,\dots, m-1\). Here, \(\overline{\phi}^{\ell}(t-k)=\left. {\phi}^{\ell}(t-k)\right|_0^\infty\). When \(\phi^\ell\) is compactly supported, so is \(\phi_{L,n}\) and the equation above implies that \(\phi_{L,n}(t)=t^m\) on \([0,1]\).NEWLINENEWLINEThe first main result states that if \( \Phi=(\phi^1,\dots,\phi^A)\) generates a multiresolution analysis for \(L^2(\mathbb{R})\) with polynomial accuracy \(m\geq 2\) with each \(\phi^i\) continuous, \(\phi^1\) supported in \([0,m]\) and \(\phi^i\) supported in \([0,1]\) for \(i=2,\dots, A\), then for \(t\in [0,1]\), \(t^{m-1}\) can be written as a linear combination of \(\phi_{L,j}\), \(j=0,\dots, m-2\), and these left edge functions together with corresponding right edge functions and \(\phi^\ell\), \(\ell=1,\dots, A\) are sufficient to construct a multiresolution analysis of \(L^2[0,1]\). As a specific case, it is shown that if \((\phi^1,\dots\phi^A)^T\) generate a multiresolution analysis of \(L^2(\mathbb{R})\) of accuracy \(A+1\), that each \(\phi^i\) is continuous and supported in \([0,2]\), and the vectors \(\{{\vec{f}}_{0,0}, {\vec{f}}_{1,0},\dots, {\vec{f}}_{A-1,0}\}\) are linearly independent then for each \(t\in [0,1]\), \(t^A\) can be expressed as a linear combination of the left edge functions \(\phi_{L,j}\), \(j=0,\dots, A-1\) and corresponding right edge functions, and \(\phi^1,\dots, \phi^A\). In particular, these functions are sufficient to generate a multiresolution analysis of \(L^2[0,1]\).NEWLINENEWLINEUnder certain extra conditions the scaling functions can be shown to be nonnegative. This requires that at least one of the components of the original scaling vector is nonnegative, and that it is possible to produce a nonnegative vector that generates the same multiresolution space which, when expanded in terms of the original scaling vectors, have coefficients satisfying a certain coercivity constraint. As with the multiresolution setup itself, the non-negativity conditions for the MRA on \([0,1]\) are inherited from corresponding non-negativity for scaling functions on the full line.