### Abstract

The subject of this paper is to study the separability of two given functions by a quadratic function. Let G be an abelian group. A function Q: G →[-∞, ∞[ is called quadratic if Q(x + y) + Q(x - y)= 2[Q(X) + Q(y)]. Let P, R: G →[ -∞, ∞[ be given functions with P ≤ Q on G. The separability of P and R by a quadratic function means that there exists a quadratic function Q satisfying P ≤ Q ≤ R on G. The following problem and its motivation comes from the theory of second-order directional derivatives: Given a function P: G → [ -∞, ∞[, we look for the existence of a family ℐ of quadratic functions such that P(x)= sup{Q(x): Q ∈ ℐ} for all x ∈ X. In order to be able to attack this problem, we shall need necessary and sufficient conditions on the quadratic separability of arbitrary functions P and R. In three sections of the paper we investigate the quadratic separability in various settings. First the case is considered when the values of P and R are finite and the underlying structure is a group. Then we deal with functions P and R that may take the value -∞. However, in this case the underlying structure is assumed to be a vector space over Q. In the third section we treat the case when topological assumptions are present, and hence the separation by continuous quadratic functions is required. The paper contains an application and examples that explain the differences between the main results of the paper.

Original language | English |
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Pages (from-to) | 209-229 |

Number of pages | 21 |

Journal | Aequationes Mathematicae |

Volume | 51 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 1996 |

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Aequationes Mathematicae*,

*51*(3), 209-229. https://doi.org/10.1007/BF01833279