Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

Abstract

It is known that, generically, Taylor series of functions holomorphic in the unit disc turn out to be universal series outside of the unit disc and in particular on the unit circle. Due to classical and recent results on the boundary behaviour of Taylor series, for functions in Hardy spaces and Bergman spaces the situation is essentially different. In this paper it is shown that in many respects these results are sharp in the sense that universality generically appears on maximal exceptional sets. As a main tool it is proved that the Taylor (backward) shift on certain Bergman spaces is mixing.

Mixing operators with prescribed unimodular eigenvalues

“…In [5] it is shown that for open sets with 0 ∈ the Taylor shift T is topologically mixing on H ( ) if and only if each connected component of C ∞ \ meets T. Results concerning topological and metric dynamics of the Taylor shift on Bergman spaces are proved in [6] and [16].…”

Banach Spaces of Universal Taylor Series in the Disc Algebra

“…So, for E = e πiC and for f in the disc algebra the only possible uniform limit function of a subsequence of (S n (f, ·)) is the function f | E and thus, in particular, A U uE = ∅. In contrast, as already mentioned in Remark 2.6, Theorem 1.1 in [5] shows that the sequence T n : H 2 → C(E) is hereditarily densely universal for E. Corollary 3.10. Let E = e iA with A ⊂ (−π, π) compact.…”

“…For m ∈ I we have e 2πiβm/β k = 1 for all k ≤ m and and thus z βm → 1 uniformly on E as m → ∞, m ∈ I.) From Theorem 1.7 and Remark 1.8 in [5] it follows that for each Dirichlet set E there exist power series having the property that (S n (ζ)) is (C, 1)-summable at all points ζ ∈ E (actually on an arc containing E) and so that {S n | E : n ∈ N 0 } is dense in C(E). Up to now, no concrete infinite set E with A U uE = ∅ is known.…”

“…If we consider the backward shift B as a mapping defined on the Hardy space H 2 instead of A(D), we obtain universality on much larger sets E. According to Carleson's theorem we have S n (f, ·) → f almost everywhere on T for all f ∈ H 2 , so we can again expect universality only on sets of vanishing measure. On the other hand, Theorem 1.1 in [5] shows that the sequence T n : H 2 → C(E) is hereditarily densely universal for all compact E of vanishing measure. If now E is a Dirichlet set, that is, for some subsequence (m k ) of the positive integers (z m k ) tends to 1 uniformly on E, then the proof of Theorem 2.5 shows that there is a residual subset of functions in…”

Universally divergent Fourier series via Landau's extremal functions

“…On other hand, recent results in [2] show that, for any compact subset E ⊆ T of arc length measure 0, the set of all f ∈ H p (D), p ∈ [1, ∞), with the property…”