# zbMATH — the first resource for mathematics

Weighted polynomial approximation in the complex plane. (English) Zbl 0963.30024
Summary: Given a pair $$(G,W)$$ of an open bounded set $$G$$ in the complex plane and a weight function $$W(z)$$ which is analytic and different from zero in $$G$$, we consider the problem of the locally uniform approximation of any function $$f(z)$$, which is analytic in $$G$$, by weighted polynomials of the form $$\left \{W^{n}(z)P_{n}(z) \right \}^{\infty}_{n=0}$$, where $$\deg P_{n} \leq n$$. The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations.
##### MSC:
 30E10 Approximation in the complex plane 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions 41A30 Approximation by other special function classes
Full Text:
##### References:
 [1] P. B. Borwein and Weiyu Chen, Incomplete rational approximation in the complex plane, Constr. Approx. 11 (1995), no. 1, 85 – 106. · Zbl 0820.41013 [2] P. B. Borwein, E. A. Rakhmanov and E. B. Saff, Rational approximation with varying weights I, Constr. Approx. 12(1996), 223-240. CMP 96:13 · Zbl 0870.41010 [3] M. von Golitschek, Approximation by incomplete polynomials, J. Approx. Theory 28 (1980), no. 2, 155 – 160. [4] A. B. J. Kuijlaars, The role of the endpoint in weighted polynomial approximation with varying weights, Constr. Approx. 12(1996), 287-301. CMP 96:13 [5] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. · Zbl 0253.31001 [6] G. G. Lorentz, Approximation by incomplete polynomials (problems and results), Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York, 1977, pp. 289 – 302. [7] G. G. Lorentz, M. von Golitschek, and Y. Makovoz, Constructive approximation, Springer-Verlag, Berlin, 1996. CMP 96:13 · Zbl 0910.41001 [8] Rolf Nevanlinna, Analytic functions, Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.12501 [9] I. E. Pritsker and R. S. Varga, The Szegö curve, zero distribution and weighted approximation, to appear in Trans. Amer. Math. Soc. CMP 96:17 · Zbl 0926.30024 [10] E. B. Saff and V. Totik, Logarithmic potentials with external fields, Springer-Verlag, Heidelberg, 1996 (to appear). · Zbl 0881.31001 [11] E. B. Saff and R. S. Varga, On incomplete polynomials, Numerische Methoden der Approximationstheorie, Band 4 (Meeting, Math. Forschungsinst., Oberwolfach, 1977) Internat. Schriftenreihe Numer. Math., vol. 42, Birkhäuser, Basel-Boston, Mass., 1978, pp. 281 – 298. [12] Vilmos Totik, Weighted approximation with varying weight, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. · Zbl 0808.41001 [13] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. [14] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Third edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1965. · Zbl 0106.28104
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.