## Abstract

Letφ:(-∞, ∞)→(0, ∞) be a given continuous even function and letmbe a positive integer. We show that, with some additional restrictions onφ, there exist decreasing sequencesx_{1}, ..., x_{m}andy_{1}, ..., y_{m-1}of symmetrically located points on (-∞, ∞) and corresponding polynomialsPandQof degreesm-1 andm, respectively, satisfyingP(x)≤φ(x)^{m},Q(x)≤φ(x) ^{m},-∞<x<∞,where equality holds with alternating signs at the corresponding sequence of points (and also at ±∞ forQ). Moreover, for any polynomialpof degree at mostm, (a)if p(x_{j})≤φ(x_{j})^{m}forj=1, ..., m, then p^{(k)}(0)≤P^{(k)}(0) wheneverkandmhave opposite parity and 0≤k<m; (b)if p(y_{j})≤φ(y_{j})^{m}forj=1, ..., m-1 and if limsup_{y→∞}p(y)/φ(y)^{m}≤1, then p^{(k)}(0)≤Q^{(k)}(0) wheneverkandmhave the same parity and 0≤k≤m. We give two computational methods for determining these sequences of points and thusPandQ.

Original language | English |
---|---|

Pages (from-to) | 293-312 |

Number of pages | 20 |

Journal | Journal of Approximation Theory |

Volume | 93 |

Issue number | 2 |

DOIs | |

State | Published - May 1998 |

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Mathematics (all)
- Applied Mathematics