Uniform convergence of polynomials associated with varying Jacobi weights. (English) Zbl 0749.41011

This article determines the functions on \([-1,1]\) that are uniform limits of weighted polynomials of the form \((1-x)^{\alpha_ n}(1+x)^{\beta_ n}p_ n(x)\), where \(\deg p_ n\leq n\), \(\lim_{n\to\infty}\alpha_ n/n=\theta_ 1\geq 0\) and \(\lim_{n\to\infty}\beta_ n/n=\theta_ 2\geq 0\). Estimates for the rate of convergence are also obtained. These results confirm a conjecture of Saff and extend previous results for incomplete polynomials.


41A10 Approximation by polynomials
Full Text: DOI


[1] E.W. Cheney, Introduction to approximation theory , McGraw-Hill, New York, 1966. · Zbl 0161.25202
[2] M.V. Golitschek, Approximation by incomplete polynomials , J. Approx. Theory 28 (1980), 155-160.
[3] M. Lachance, E.B. Saff and R.S. Varga, Bounds for incomplete polynomials vanishing at both endpoints of an interval , in Constructive approaches to mathematical models (C.V. Coffman, and G.J. Fix, eds.), Academic Press, New York, 1979, 421-437. · Zbl 0441.41004
[4] G.G. Lorentz, Approximation by incomplete polynomials ( problems and results ), in PadĂ© and rational approximation: Theory and applications (E.B. Saff and R.S. Varga, eds.), Academic Press, New York, 1977, 289-302. · Zbl 0383.41004
[5] D.S. Lubinsky, H.N. Mhaskar and E.B. Saff, A proof of Freud’s conjecture for exponential weights , Constr. Approx. 4 (1988), 65-83. · Zbl 0653.42024
[6] ——– and E.B. Saff, Uniform and mean approximation, by certain weighted polynomials with applications , Constr. Approx. 4 (1988), 21-64. · Zbl 0646.41003
[7] ——– and ——–, Strong asymptotics for extremal errors and extremal polynomials associated with weights on \((-\infty,\infty)\) , Lecture Notes in Mathematics, vol. 1305-1988. · Zbl 0647.41001
[8] H.N. Mhaskar and E.B. Saff, Where does the sup norm of a weighted polynomial live? , Constr. Approx. 1 (1985), 71-91. · Zbl 0582.41009
[9] ——– and ——–, A Weierstrass-type theorem for certain weighted polynomials , in Approximation theory and applications (S.P. Singh, ed.), Pitman Publishing Ltd, 1985, 115-123. · Zbl 0578.41007
[10] E.B. Saff, J.L. Ullman and R.S. Varga, Incomplete polynomials: an electrostatics approach , in Approximation theory III (E.W. Cheney, ed.), Academic Press, New York, 1980, 769-782. · Zbl 0479.41016
[11] E.B. Saff and R.S. Varga, Uniform approximation by incomplete polynomials , Internat. J. Math. Math. Sci. 1 (1978), 407-420. · Zbl 0421.41006
[12] J.L. Walsh, Interpolation and approximation by rational functions in the complex domain , Colloquium Publication, XX, 5th Ed., Amer. Math. Soc., Providence, Rhode Island, 1969.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.