Pathological function spaces and an unsolved Analysis I problem

Abstract

The spaces in the title are holomorphic function spaces on the unit disk containing a function with the bizarre property that, although it can be approximated in norm by polynomials, it cannot be approximated by the partial sums of its own Taylor series, nor by any linear combinations thereof. The search for concrete examples of such spaces leads to an elementary problem about convergent series that is still unsolved.

18 pages