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AMBIENTUM BIOETHICA BIOLOGIA CHEMIA DIGITALIA DRAMATICA EDUCATIO ARTIS GYMNAST. ENGINEERING EPHEMERIDES EUROPAEA GEOGRAPHIA GEOLOGIA HISTORIA HISTORIA ARTIUM INFORMATICA IURISPRUDENTIA MATHEMATICA MUSICA NEGOTIA OECONOMICA PHILOLOGIA PHILOSOPHIA PHYSICA POLITICA PSYCHOLOGIA-PAEDAGOGIA SOCIOLOGIA THEOLOGIA CATHOLICA THEOLOGIA CATHOLICA LATIN THEOLOGIA GR.-CATH. VARAD THEOLOGIA ORTHODOXA THEOLOGIA REF. TRANSYLVAN
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STUDIA MATHEMATICA - Issue no. 2 / 2015 | |||||||
Article: |
OPTIMAL CUBIC LAGRANGE INTERPOLATION: EXTREMAL NODE SYSTEMS WITH MINIMAL LEBESGUE CONSTANT. Authors: . |
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Abstract: In the theory of interpolation of continuous functions by algebraic polynomials of degree at most n − 1≥ 2, the search for explicit analytic expressions of extremal node systems which lead to the minimal Lebesgue constant is still anintriguing topic in mathematics today [33]. The first non-trivial case n − 1 = 2(quadratic interpolation) has been completely resolved, even in two alternative fashions, see [25], [27]. In the present paper we proceed to completely resolve thecubic case (n − 1 = 3) of optimal polynomial Lagrange interpolation on the unit interval [−1, 1]. We will provide two explicit analytic expressions for the uncountable infinitely many extremal node systems x*1 < x*2 < x*3 < x*4 in [−1, 1] which all lead to the (known) minimal Lebesgue constant of cubic Lagrange interpolationon [−1, 1]. The descriptions of the extremal node systems (which need notbe zero-symmetric) resemble the solutions for the quadratic case and incorporate two intrinsic constants expressed by radicals, of which one constant looks particularly intricate. Our results encompass earlier related work provided in [17], [23],[24], [29], [30] and are guided by symbolic computation. Mathematics Subject Classification (2010): 05C35, 33F10, 41A05, 41A44, 65D05,68W30. Keywords: Constant, cubic, extremal, interpolation, Lagrange interpolation, Lebesgue constant, minimal, node, node system, optimal, point, polynomial, symboliccomputation.
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