Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant
Keywords:
Constant, cubic, extremal, interpolation, Lagrange interpolation, Lebesgue constant, minimal, node, node system, optimal, point, polynomial, symbolic computation.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 an intriguing 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 the cubic 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 interpolation on [−1, 1]. The descriptions of the extremal node systems (which need not be 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.References
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