Spurious limit functions of Taylor series

It is known that, generically in the space H(𝔻) of functions holomorphic in the unit disc 𝔻 , the sequences (S_n f) of partial sums of Taylor series behave extremely erratically on the unit circle 𝕋 . According to a result of Gardiner and Manolaki, the situation changes in a significant way if f ∈ H(𝔻) has nontangential limits on subsets of 𝕋 of positive arc length measure. In this case each convergent subsequence tends to the nontangential limit function almost everywhere. We consider the question to which extent in spaces of holomorphic functions where nontangential limits are guaranteed, “spurious” limit functions, that is, limit functions different than the nontangential limit may appear on small subsets of 𝕋 .