Polynomial density in Bers spaces
Author:
Jacob Burbea
Journal:
Proc. Amer. Math. Soc. 62 (1977), 8994
MSC:
Primary 30A98
MathSciNet review:
0425139
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Abstract: Let D be a bounded Jordan domain such that for . Here is the Poincaré metric for D. Define , the Bers space, to be the Fréchet space of holomorphic functions f on D, such that is finite, . It is well known that the polynomials are dense in for . We show that they are dense in for irrespective whether the boundary of D is rectifiable or not.
 [1]
Lipman
Bers, Automorphic forms and Poincaré series for infinitely
generated Fuchsian groups, Amer. J. Math. 87 (1965),
196–214. MR 0174737
(30 #4937)
 [2]
Lipman
Bers, A nonstandard integral equation with applications to
quasiconformal mappings, Acta Math. 116 (1966),
113–134. MR 0192046
(33 #273)
 [3]
Peter
L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and
Applied Mathematics, Vol. 38, Academic Press, New YorkLondon, 1970. MR 0268655
(42 #3552)
 [4]
C.
J. Earle and A.
Marden, Projections to automorphic
functions, Proc. Amer. Math. Soc. 19 (1968), 274–278. MR 0224813
(37 #412), http://dx.doi.org/10.1090/S00029939196802248136
 [5]
G.
H. Hardy and J.
E. Littlewood, Some properties of fractional integrals. II,
Math. Z. 34 (1932), no. 1, 403–439. MR
1545260, http://dx.doi.org/10.1007/BF01180596
 [6]
Lars
Inge Hedberg, Weighted mean square approximation in plane regions,
and generators of an algebra of analytic functions, Ark. Mat.
5 (1965), 541–552 (1965). MR 0219729
(36 #2808)
 [7]
Marvin
I. Knopp, A corona theorem for automorphic forms and related
results, Amer. J. Math. 91 (1969), 599–618. MR 0251219
(40 #4450)
 [8]
Thomas
A. Metzger, On polynomial approximation in
𝐴_{𝑞}(𝐷), Proc. Amer.
Math. Soc. 37
(1973), 468–470. MR 0310260
(46 #9361), http://dx.doi.org/10.1090/S00029939197303102607
 [9]
Thomas
A. Metzger, On polynomial density in
𝐴_{𝑞}(𝐷), Proc. Amer.
Math. Soc. 44
(1974), 326–330. MR 0340623
(49 #5375), http://dx.doi.org/10.1090/S00029939197403406236
 [10]
Mark
Sheingorn, Poincaré series of polynomials bounded away from
zero on a fundamental region, Amer. J. Math. 95
(1973), 729–749. MR 0344455
(49 #9194)
 [11]
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. MR 0218588
(36 #1672b)
 [1]
 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)
 [2]
 , A nonstandard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113134. MR 33 #273. MR 0192046 (33:273)
 [3]
 P. L. Duren, Theory of spaces, Pure and Appl. Math., vol. 38, Academic Press, New York and London, 1970. MR 42 #3552. MR 0268655 (42:3552)
 [4]
 C. J. Earle and A. Marden, Projections to automorphic functions, Proc. Amer. Math. Soc. 19 (1968), 274278. MR 37 #412. MR 0224813 (37:412)
 [5]
 G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1931), 403439. MR 1545260
 [6]
 L. I. Hedberg, Weighted mean square approximation in plane regions, and generators of an algebra of analytic functions, Ark. Mat. 5 (1965), 541552. MR 36 #2808. MR 0219729 (36:2808)
 [7]
 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)
 [8]
 T. A. Metzger, On polynomial approximation in , Proc. Amer. Math. Soc. 37 (1973), 468470. MR 46 #9361. MR 0310260 (46:9361)
 [9]
 , On polynomial density in , Proc. Amer. Math. Soc. 44 (1974), 326330. MR 49 #5375. MR 0340623 (49:5375)
 [10]
 M. Sheingorn, Poincaré series of polynomials bounded away from zero on fundamental region, Amer. J. Math. 95 (1973), 729749. MR 49 #9194. MR 0344455 (49:9194)
 [11]
 J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence, R. I., 1935. MR 0218588 (36:1672b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197704251393
PII:
S 00029939(1977)04251393
Keywords:
Bers spaces,
Poincaré metric,
polynomial density
Article copyright:
© Copyright 1977
American Mathematical Society
