On polynomial density in
Author:
Thomas A. Metzger
Journal:
Proc. Amer. Math. Soc. 44 (1974), 326330
MSC:
Primary 30A98
MathSciNet review:
0340623
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Abstract: Let be a bounded Jordan domain. Define , the Bers space, to be the Banach space of holomorphic functions on , such that is finite, where is the Poincaré metric for . It is well known that the polynomials are dense in for and we shall prove they are dense in for if the boundary of is rectifiable. Also some remarks are made in case the boundary of is not rectifiable.
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 D. Aharonov, A. Shields and H. Shapiro, Weakly invertible elements in the space of squaresummable holomorphic functions (to appear). MR 0365150 (51:1403)
 [2]
 L. Bers, Automorphic forms and Poincaré series for infinitely generated Fuchsian groups, Amer. J. Math. 87 (1965), 196214. MR 30 #4937. MR 0174737 (30:4937)
 [3]
 , A nonstandard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113134. MR 33 #273. MR 0192046 (33:273)
 [4]
 P. L. Duren, Theory of spaces, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR 42 #3552. MR 0268655 (42:3552)
 [5]
 C. J. Earle and A. Marden, Projections to automorphic functions, Proc. Amer. Math. Soc. 19 (1968), 274278. MR 37 #412. MR 0224813 (37:412)
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 T. Gamelin, spaces and extremal functions in , Trans. Amer. Math. Soc. 124 (1966), 158167. MR 35 #4731. MR 0213877 (35:4731)
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 M. I. Knopp, A corona theorem for automorphic forms and related results, Amer. J. Math. 91 (1969), 599618. MR 40 #4450. MR 0251219 (40:4450)
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 T. A. Metzger and K. V. Rajeswara Rao, On integrable and bounded automorphic forms, Proc. Amer. Math. Soc. 28 (1971), 562566. MR 43 #6432. MR 0280713 (43:6432)
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 T. A. Metzger and M. Sheingorn, Polynomial approximation in the Bers' spaces (to appear).
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 T. A. Metzger, On polynomial approximation in , Proc. Amer. Math. Soc. 37 (1973), 468470. MR 0310260 (46:9361)
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 M. Sheingorn, Poincaré series of polynomials bounded away from zero on a fundamental region, Amer. J. Math. (to appear). MR 0344455 (49:9194)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197403406236
PII:
S 00029939(1974)03406236
Keywords:
Polynomial density,
Bers spaces
Article copyright:
© Copyright 1974
American Mathematical Society
