The Convergence of Polynomial Interpolation and Runge Phenomenon
Keywords:
Polynomial interpolation, Lagrange polynomial, Chebyshev polynomial, Chebyshev pointsAbstract
One of the main questions is whether or not a sequence of polynomials pn(x) that interpolate a continuous function f at n + 1 equally spaced points tends to f in the sup-norm? The answer is "no" in some cases. The main fact is that interpolant polynomials pn(x) of a function f converge at a rate determined by the smoothness of f: the pn(x) converge rapidly to the function f if it is k-times differentiable and converges exponentially if f is analytic. The polynomial interpolation depends on n but it also depends on the way in which the points are distributed. We determine conditions on the function f to ensure the convergence of the polynomials pn(x) to the function f, as the continuity of the function is not enough. The question for analytic functions is answered using potential theory. Convergence and divergence rate of interpolants of analytic functions on the interval are investigated. We also study a generalized Runge phenomenon and find out how the location of the points and poles affect the convergence.
References
Taylor W. Method of Lagrangian curvilinear interpolation. Journal of Research of the National Bureau of Standard, 35, 1945.
Natanson I, Obolensky A and Schulenberger J. Constructive Function Theory. volume 1.Ungar New York, 1964.
Davis P. Interpolation and Approximation . Blaisdell Publishing Company,1965.
Fox L and Parker T. Chebyshev Polynomials in Numerical Analysis. Oxford Press, 1968.
Salzer H. Langrangian Interpolation at Chebyshev Points. The Computer Journal , 1972.
Rivlin T. The Chebyshev Polynomials. A Wiley-Interscience publication, 1974.
Epperson J. On the Runge Example. Amer Math,94(4), 1987.
Ransford T. Potential Theory in Complex Plane. Cambridge University Press, 1995.
Mason J and Handscomb D. Chebyshev Polynomials. CRC Press, 2003.
Berrut J and Trefethen L. Barycentric Lagrange Interpolation. SIAM Review, 2004.
Powell M. Approximation Theory and Methods. Cambridge University Press, 2004.
Higham N. The Numerical Stabilty of Barycentric Lagrange Interpolation. IMA Journal of Numerical Analysis, 2004.
Trefethen L. Six Myths of Polynomial Interpolation and Quadrature. Maths. Today 47, 2011.
Platte R, Trefethen, L. and Kuijlaars A. Impossibility of Approximating Analytic Functions from Equispaced Samples. SIAM Journal, 2011.
Trefethen N. Approximation Theory and Approximation Practice. University of Oxford, 2013.
