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    STUDIA MATHEMATICA - Issue no. 2 / 2015  
         
  Article:   OPTIMAL CUBIC LAGRANGE INTERPOLATION: EXTREMAL NODE SYSTEMS WITH MINIMAL LEBESGUE CONSTANT.

Authors:  . 		 
       
         
  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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