Approximation on disks

de Paepe, P. J.

doi:10.1090/S0002-9939-1986-0835885-0

Approximation on disks
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Proc. Amer. Math. Soc. 97 (1986), 299-302 Request permission

Abstract:

It is shown that if the functions $F$ and $G$ are defined in a neighborhood of the origin in the complex plane and are in a certain sense like ${z^m}$ and ${z^{ - n}}$ with $\gcd (m,n) = 1$, then on sufficiently small closed disks $D$ around 0 every continuous function on $D$ can be uniformly approximated by polynomials in $F$ and $G$.

References

Michael Freeman, The polynomial hull of a thin two-manifold, Pacific J. Math. 38 (1971), 377–389. MR 308442, DOI 10.2140/pjm.1971.38.377

Uniform approximations to function of a complex variable

3

Steven Minsker, Some applications of the Stone-Weierstrass theorem to planar rational approximation, Proc. Amer. Math. Soc. 58 (1976), 94–96. MR 467322, DOI 10.1090/S0002-9939-1976-0467322-6

Approximation by polynomials in two complex variables

246

A. G. O’Farrell and K. J. Preskenis, Approximation by polynomials in two diffeomorphisms, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 1, 105–107. MR 722862, DOI 10.1090/S0273-0979-1984-15203-0
P. J. de Paepe, Some applications of the Stone-Weierstrass theorem, Proc. Amer. Math. Soc. 70 (1978), no. 1, 63–66. MR 493360, DOI 10.1090/S0002-9939-1978-0493360-5
Kenneth John Preskenis, Approximation on disks, Trans. Amer. Math. Soc. 171 (1972), 445–467. MR 312123, DOI 10.1090/S0002-9947-1972-0312123-3
Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083
J. Wermer, Polynomially convex disks, Math. Ann. 158 (1965), 6–10. MR 174968, DOI 10.1007/BF01370392