Arkiv för Matematik

  • Ark. Mat.
  • Volume 34, Number 1 (1996), 103-117.

Distribution of interpolation points

René Grothmann

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We show that interpolation to a function, analytic on a compact set E in the complex plane, can yield maximal convergence only if a subsequence of the interpolation points converges to the equilibrium distribution on E in the weak sense. Furthermore, we will derive a converse theorem for the case when the measure associated with the interpolation points converges to a measure on E, which may be different from the equilibrium measure.

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Ark. Mat., Volume 34, Number 1 (1996), 103-117.

Received: 7 August 1995
First available in Project Euclid: 31 January 2017

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1996 © Institut Mittag-Leffler


Grothmann, René. Distribution of interpolation points. Ark. Mat. 34 (1996), no. 1, 103--117. doi:10.1007/BF02559510.

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  • Blatt, H.-P., On the distribution of simple zeros of polynomials, J. Approx. Theory 69 (1992), 250–268.
  • Blatt, H.-P., The distribution of sign changes of the error function in best Lp-approximation, in Computational Methods and Function Theory (Ali, R. M., Ruscheweyh, St. and Saff, E. B., eds.), pp. 75–87, World Scientific Publishing, London, 1995.
  • Carleson, L., Mergelyan's theorem on uniform polynomial approximation, Math. Scand. 15 (1964), 167–175.
  • Deny, J., Systèmes totaux de fonctions harmoniques. Ann. Inst. Fourier (Grenoble) 1 (1950), 103–113.
  • Grothmann, R., On the real CF-approximation for polynomial approximation and strong uniqueness constants, J. Approx. Theory 55 (1988), 86–103.
  • Kadec, M. I., On the distribution of points of maximal deviation in the approximation of continuous functions by polynomials, Uspekhi Mat. Nauk 15 (1960), 199–202. English transl.: Trans. Amer. Math. Soc. 26 (1960), 231–234.
  • Krylow, V. J., Approximate Calculation of Integrals Macmillan Company, New York, 1962.
  • Saff, E. B. and Shekhtman, B., Interpolatory properties of best L2-approximants, Indag. Math. (N. S.) 1 (1990), 489–498.
  • Tsuji, M., Potential Theory in Modern Function Theory, Maruzen Co. Ltd., Tokyo, 1959. Also: Chelsea Publ. Co., New York, 1975.
  • Wallin, H., Rational interpolation to meromorphic functions, in Padé Approximation and its Applications, Proc. Conf. Amsterdam 1980, (Bruin, M. G. and van Rossum, H., eds.), Lecture Notes in Math. 888, pp. 371–383, Springer-Verlag, Berlin-Heidelberg, 1981.
  • Wallin, H., Divergence of multipoint Padé approximation, in Complex Analysis, Proc. 5th Rom.-Finn. Seminar Bukarest 1981, Part 2, (Cazacu, A., Boboc, N. and Jurchescu, M., eds.), Lecture Notes in Math. 1014, pp. 246–255, Springer-Verlag, Berlin-Heidelberg, 1983.
  • Wallin, H., Convergence and divergence of multipoint Padé approximants of meromorphic functions, in Rational Approximation and Interpolation, Proc. Conf. Tampa (Graves-Morris, P. R., Saff, E. B. and Varga, R. S., eds.), Lecture Notes in Math. 1105, pp. 272–284, Springer-Verlag, Berlin-Heidelberg, 1984.
  • Walsh, J. L., Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Colloq. 20, Amer. Math. Soc., Providence, R. I., 1935.