Approximation on disks

Author:
P. J. de Paepe

Journal:
Proc. Amer. Math. Soc. **97** (1986), 299-302

MSC:
Primary 30E10; Secondary 46J10

DOI:
https://doi.org/10.1090/S0002-9939-1986-0835885-0

MathSciNet review:
835885

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if the functions and are defined in a neighborhood of the origin in the complex plane and are in a certain sense like and with , then on sufficiently small closed disks around 0 every continuous function on can be uniformly approximated by polynomials in and .

**[1]**M. Freeman,*The polynomial hull of a thin two-manifold*, Pacific J. Math.**38**(1971), 377-389. MR**0308442 (46:7556)****[2]**S. N. Mergelyan,*Uniform approximations to function of a complex variable*, Amer. Math. Soc. Tralsi. (1)**3**(1962), 281-391.**[3]**S. Minsker,*Some applications of the Ston- Weierstrass theorem to planar rational approximation*, Proc. Amer. Math. Soc.**58**(1976), 94-96. MR**0467322 (57:7181)****[4]**A. G. O'Farrell and K. J. Preskenis,*Approximation by polynomials in two complex variables*, Math. Ann.**246**(1980), 225-232.**[5]**-,*Approximation by polynomials in two diffeomorphisms*, Bull. Amer. Math. Soc. (N.S.)**10**(1984), 105-107. MR**722862 (85b:30053)****[6]**P. J. de Paepe,*Some applications of the Ston- Weierstrass theorem*, Proc. Amer. Math. Soc.**70**(1978), 63-66. MR**0493360 (58:12385)****[7]**K. J. Preskenis,*Approximation on disks*, Trans. Amer. Math. Soc.**171**(1972), 445-467. MR**0312123 (47:685)****[8]**E. L. Stout,*The theory of uniform algebras*, Bogden and Quigley, Tarrytown-on-Hudson, N. Y., 1971. MR**0423083 (54:11066)****[9]**J. Wermer,*Polynomially convex disks*, Math. Ann.**158**(1965), 6-10. MR**0174968 (30:5158)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0835885-0

Keywords:
Function algebra,
Stone-Weierstrass theorem,
uniform approximation in the complex plane

Article copyright:
© Copyright 1986
American Mathematical Society