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Using a recent Mergelyan type theorem for products of planar compact sets we establish generic existence of Universal Taylor Series on products of planar simply connected domains Ω i , i = 1, . . . , d. The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets K i , i = 1, . . . , d such that C − K i is connected and for at least one i 0 the set K i0 is disjoint from Ω i0 .

We consider Bergman spaces and variations of them on domains  in one or several complex variables. For certain domains  we show that the generic function in these spaces is totaly unbounded in  and hence non-extendable. We also show that generically these functions do not belong -not even locally -in Bergman spaces of higher order. Finally, in certain domains  , we give examples of bounded non-extendable holomorphic functions -a generic result in the spaces ) (  s of holomorphic functions in  whose derivatives of order s  extend continuously to  (    s 0 ). AMS classification numbers:Primary 30H20, 32A36, Secondary 32D05, 32T05, 54E52.

In this paper we prove generic results concerning Hardy spaces in one or several complex variables. More precisely, we show that the generic function in certain Hardy type spaces is totally unbounded and hence non-extentable, despite the fact that these functions have non-tangential limits at the boundary of the domain. We also consider local Hardy spaces and show that generically these functions do not belongnot even locallyto Hardy spaces of higher order. We work first in the case of the unit ball of n C where the calculations are easier and the results are somehow better, and then we extend them to the case of strictly pseudoconvex domains. ) (B which are totally unbounded in B is dense and  G in this space.

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