Towards Best Approximations for \\(\\left \\vert x\\right \\vert ^{\\alpha }\\)

Abstract

In this survey, we discuss asymptotic relations for the approximation of \\(\\left \\vert x\\right \\vert ^{\\alpha },\\alpha >0\\) in \\(L_{\\infty }\\left [ -1,1\\right ] \\) by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind. Along the way, we explore connections of our results together with papers of Ganzburg (J Approx Theory 119:193–213, 2002) and Lubinsky (Constr Approx 25(3):303–366, 2007), by presenting numerical results, indicating a possible constructive way towards some representations for the Bernstein constants.