Universal Functions on Complex General Linear Groups |
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Authors: | Yukitaka Abe Paolo Zappa |
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Institution: | a Department of Mathematics, Toyama University, Toyama, 930-8555, Japan;b Dipartimento di Matematica, Università di Perugia, 06123, Perugia, Italy |
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Abstract: | In 1929, Birkhoff proved the existence of an entire function F on
with the property that for any entire function f there exists a sequence {ak} of complex numbers such that {F(ζ+ak)} converges to f (ζ) uniformly on compact sets. Luh proved a variant of Birkhoff's theorem and the second author proved a theorem analogous to that of Luh for the multiplicative group
*. In this paper extensions of the above results to the multi-dimensional case are proved. Let M(n,
) be the set of all square matrices of degree n with complex coefficients, and let G=GL(n,
) be the general linear group of degree n over
. We denote by
(G) the set of all holomorphic functions on G. Similarly, we define
(
). Let K be the
(G)-hull of a compact set K in G. Finally we denote by B(G) the set of all compact subsets K of G with K=K such that there exists a holomorphic function f on M(n,
) with f(0)(f(K)), where (f(K)) is the
(
)-hull of f(K). Our main result is the following. There exists a holomorphic function F on G such that for any KB(G), for any function f holomorphic in some neighbourhood of K, and for any >0, there exists CG with maxZK |F(CZ)−f(Z)|<. |
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Keywords: | |
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