Perfect splines with boundary conditions of least norm (Q1329030)

The author adresses a certain minimal norm problem for perfect splines. To state his main result, we first have to introduce some notation: Let there be given matrices \(A = ((a_{ij}))\) and \(B = ((b_{ij}))\), such that \(((( - 1)^ j a_{ij}))\) as well as \(B\) are sign consistent. Furthermore, with numbers \(\nu_ i\), \(i = 1, \dots, n\) such that \(1 \leq \nu_ i \leq r\) for all \(i\), let \(P_{rN} (\nu_ 1, \dots, \nu_ n)\) denote the space of all perfect splines of degree \(r\) on \([0,1]\) with \(N\) inner knots, having \(n\) distinct zeros of multiplicity \(\nu_ i\) respectively in \((0,1)\). Then the following result holds: There is a uniquely determined \(P \in P_{rN} (\nu_ 1, \dots, \nu_ n)\) satisfying \[ A \bigl( P(0), P'(0), \dots, P^{(r - 1)} (0) \bigr)^ T = 0 \quad \text{and} \quad B \bigl( P(1), P'(1), \dots, P^{(r - 1)}(1) \bigr)^ T = 0, \tag{1} \] which minimizes the uniform norm \(\| p \|_{[0,1]}\) under all \(p \in P_{rN} (\nu_ 1, \dots, \nu_ n)\) satisfying (1). Moreover, \(P\) is characterized by a certain alternation property. The paper consists mainly in the proof of this result.