Approximation on disks

Author:
Kenneth John Preskenis

Journal:
Trans. Amer. Math. Soc. **171** (1972), 445-467

MSC:
Primary 41A20; Secondary 30A82

DOI:
https://doi.org/10.1090/S0002-9947-1972-0312123-3

MathSciNet review:
0312123

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Abstract: Let be a closed disk in the complex plane, a complex valued continuous function on and the uniform closure on of rational functions in and which are finite. Among other results we obtain the following. Theorem. *If is of class in a neighborhood of and everywhere (i.e., is an orientation reversing immersion of in the plane), then* . Theorem. *Let be a polynomial in and . If for each a in with at he zeros of in where , then* . Corollary. *Let be a polynomial in and and let . Then there exists an such that, for* . The proofs of the theorems use measures and the conditions involved in the theorems are independent of each other. Concerning the corollary, results of E. Bishop and G. Stolzenberg show that and , then there exists no such that where .

Let be a map on = unit polydisk in with values in = uniform closure on of polynomials in . Theorem. *If is of class in a neighborhood of is invertible and if for each in , there exist complex constants , such that has positive real part for all , then is a polynomially convex set*. Corollary. *If where and the coefficients satisfy and , then* . Corollary. *If where is of class and satisfies the Lipschitz condition with , then* . This last corollary is a result of Hörmander and Wermer. The proof of the theorem uses methods from several complex variables.

**[1]**E. Bishop,*A minimal boundary for function algebras*, Pacific J. Math. 9 (1959), 629-642. MR 22 #191. MR**0109305 (22:191)****[2]**-,*Boundary measure of analytic differentials*, Duke Math. J. 27 (1960), 331-340. MR 22 #9621. MR**0118852 (22:9621)****[3]**A. Browder,*Introduction to function algebras*, Benjamin, New York, 1969. MR 39 #7431. MR**0246125 (39:7431)****[4]**L. De Branges,*The Stone-Weierstrass theorem*, Proc. Amer. Math. Soc. 10 (1959), 822-824. MR 22 #3970. MR**0113131 (22:3970)****[5]**M. Freeman,*Some conditions for uniform approximation on a manifold*, Proc. Internat. Sympos. on Function Algebras (Tulane Univ., 1965), Scott-Foresman, Chicago, Ill., 1966, pp. 42-60. MR 33 #1758. MR**0193538 (33:1758)****[6]**R. Gunning and H. Rossi,*Analytic functions of several complex variables*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 31 #4927. MR**0180696 (31:4927)****[7]**L. Hörmander,*An introduction to complex analysis in several variables*, Van Nostrand, Princeton, N. J., 1966. MR 34 #2933. MR**0203075 (34:2933)****[8]**L. Hörmander and J. Wermer,*Uniform approximation on compact sets in*, Math. Scand. 23 (1968), 5-21. MR 40 #7484. MR**0254275 (40:7484)****[9]**J. J. Kohn,*Harmonic integrals on strongly pseudo-convex manifolds*. I, II, Ann. of Math. (2) 78 (1963), 112-148; ibid. (2) 79 (1964), 450-472. MR 27 #2999; MR 34 #8010. MR**0153030 (27:2999)****[10]**M. Kreĭn and D. Milman,*On extreme points of regular convex sets*, Studia. Math. 9 (1940), 133-138. MR 3, 90. MR**0004990 (3:90a)****[11]**S. N. Mergelian,*Uniform approximations to functions of a complex variable*, Uspehi Mat. Nauk 5 (1952), no. 2 (48), 31-122; English transl., Amer. Math. Soc. Transl. (1) 3 (1962), 294-391. MR 14, 547.**[12]**R. Nirenberg and R. O. Wells, Jr.,*Holomorphic approximation on real submanifolds of a complex manifold*, Bull. Amer. Math. Soc. 73 (1967), 378-381. MR 35 #746. MR**0209850 (35:746)****[13]**K. Oka,*Domaines convexes pat rapport aux fonctions rationelles*, J. Sci. Hiroshima Univ. 6 (1936), 245-255.**[14]**K. Preskenis,*On a theorem of S. N. Mergelyan*, Master's Thesis, Brown University, Providence, R. I., 1967.**[15]**W. Rudin,*Real and complex analysis*, McGraw-Hill, New York, 1966. MR 35 #1420. MR**0210528 (35:1420)****[16]**G. Stolzenberg,*Polynomially and rationally convex sets*, Acta. Math. 109 (1963), 259-289. MR 26 #3929. MR**0146407 (26:3929)****[17]**F. A. Valentine,*A Lipschitz condition preserving extension for a vector function*, Amer. J. Math. 67 (1945), 83-93. MR 6, 203. MR**0011702 (6:203e)****[18]**J. Wermer,*Approximation on a disk*, Math. Ann. 155 (1964), 331-333. MR 29 #2670. MR**0165386 (29:2670)****[19]**-,*Polynomially convex disks*, Math. Ann. 158 (1965), 6-10. MR 30 #5158. MR**0174968 (30:5158)**

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DOI:
https://doi.org/10.1090/S0002-9947-1972-0312123-3

Article copyright:
© Copyright 1972
American Mathematical Society