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  Volume 10, Issue 2, Article 33
On a Weighted Interpolation of Functions with Circular Majorant

    Authors: Simon J. Smith,  
    Keywords: Interpolation, Lagrange interpolation, Weighted interpolation, Circular majorant, Projection norm, Lebesgue constant, Chebyshev polynomial.  
    Date Received: 31/07/08  
    Date Accepted: 06/02/09  
    Subject Codes:

41A05, 41A10.

    Editors: Qazi Ibadur Rahman,  

Denote by the projection operator obtained by applying the Lagrange interpolation method, weighted by , at the zeros of the Chebyshev polynomial of the second kind of degree . The norm , where denotes the supremum norm on , is known to be asymptotically the same as the minimum possible norm over all choices of interpolation nodes for unweighted Lagrange interpolation. Because the projection forces the interpolating function to vanish at , it is appropriate to consider a modified projection norm , where is a given function (a curved majorant) that satisfies and . In this paper the asymptotic behaviour of the modified projection norm is studied in the case when is the circular majorant . In particular, it is shown that asymptotically is smaller than by the quantity .

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