A simple proof of the Gasca-Maeztu conjecture for ɳ = 4

• Autores: V. Bayramyan, H. Hakopian, S. Toroyan
• Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 7, Nº. 1, 2015, págs. 137-147
• Idioma: inglés
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• Resumen

  • In this paper we consider ɳ -poised node sets whose all ɳ-fundamental polynomials are products of ɳ linear factors, as it always takes place in the univariate case. Gasca and Maeztu conjectured [6] that every such set possesses a maximal line, i.e., a line passing through its ƞ + 1 nodes. Till now the conjecture is confirmed to be true for ɳ ≤ 5. The case ɳ = 4 was proved for the first time by Busch [2]. The case ƞ = 5 was proved recently by Hakopian, Jetter and Zimmermann [9]. In this paper we bring a simple and short proof of the conjecture for ɳ = 4. In the proof we use some new ideas and methods of the above mentioned proof for the case ɳ = 5.