Mini-Workshop: Complex Approximation and Universality

The notion of universality covers a wide range of phenomena in complex analysis. Generally speaking, a universal object is one which, when subjected to some limiting process, approximates every object in some universe. For example, universality occurs when the translates of an entire function can approximate any other entire function, or when the partial sums of a formal power series or a formal trigonometric series approximate all functions in some natural class. For a long time, existing approximation theorems were used in constructions of universal functions and universal series. In recent years, however, constructions have required the development of new approximation theorems, thereby also enriching the area of complex approximation. Universal functions. There is no single definition of a universal function. What they have in common is the following. One considers a suitable sequence \mathcal T = (T_n) of operators acting on a space X , for example, of holomorphic functions with values in another space Y of holomorphic functions. Then a function f\in X is called universal with respect to \mathcal T if the sequence (T_nf) is dense in Y . One of the earliest examples of a universal function is due to Birkhoff (1929) who showed that there exists an entire function f whose translates f(z+n), n \ge 1, can approximate any other entire function, uniformly on compact sets. In that case we have (T_nf)(z) = f(z +n), and X = Y is the space of entire functions with the usual compact-open topology. Seidel and Walsh showed that an analogue of Birkhoff's universality theorem holds for functions holomorphic in the unit disc, if we replace translates by “non-euclidian translates”, that is T_nf=f\circ\phi_n is the composition of f with an automorphism \phi_n of the unit disc D . At the heart of the study of holomorphic functions in the disc D is the class H^\infty(D) of bounded holomorphic functions on the disc. Chee showed the existence of universal functions for the class H^\infty(B) of bounded holomorphic functions on the unit ball of C^N . Richard Aron's talk was concerned with the size and the structure of the set of such universal functions. In the study of the space H^\infty(B) a fundamental role is played by inner functions. These are also of importance in engineering control theory. Recently, Gauthier and Xiao have shown the existence of universal inner functions in the unit ball of C^N . Geir Arne Hjelle and Raymond Mortini gave talks concerned with approximating inner functions in the unit disc D by simpler inner functions, namely Blaschke products. Extending the study of functions in the unit disc, which are universal with respect to composition with automorphisms of the disc, Mortini talked about the universality of functions f holomorphic on a domain \Omega with respect to a sequence (f\circ\phi_n) of compositions, where (\phi_n) are self-maps of \Omega (not necessarily automorphisms). Universal series. In 1918 Jentzsch gave an example of a power series \Sigma for which a subsequence of the partial sums of \Sigma converges outside of its disc D of convergence. Such a power series is said to be overconvergent. Luh, Chui and Parnes showed the existence of such an overconvergent series \Sigma which is universal in the sense that, for each compact set K in the complement of \overline D , and for each f holomorphic on K , there are partial sums of \Sigma which converge uniformly to f . Nestoridis showed that one can even allow K to meet the boundary of D

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