## Abstract

Consider a spline s(x) of degree n with L knots of specified multiplicities R_{1}, ..., R_{L}, which satisfies r sign consistent mixed boundary conditions in addition to s^{(n)}(a) = 1. Such a spline has at most n + 1 -r + ∑_{j = 1}^{L} R_{j} zeros in (a, b) which fulfill an interlacing condition with the knots if s(x) ≢ = 0 everywhere. Conversely, given a set of n -r + ∑_{j = 1}^{L} R_{j} zeros then for any choice η_{1} < ··· < η_{L} of the knot locations which fulfills the interlacing condition with the zeros, the unique spline s(x) possessing these knots and zeros and satisfying the boundary conditions is such that s^{(n)}(x) vanishes nowhere and changes sign at η_{j} if and only if R_{j} is odd. Moreover there exists a choice of the knot locations, not necessarily unique, which makes |s^{(n)}(x)| ≡ 1. In particular, this establishes the existence of monosplines and perfect splines with knots of given multiplicities, satisfying the mixed boundary conditions and possessing a prescribed maximal zero set. An application is given to double-precision quadrature formulas with mixed boundary terms and a certain polynomial extremal problem connected with it.

Original language | English |
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Pages (from-to) | 739-751 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 61 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 1977 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics