The Fourier series of the elements in the generalized Bergman spaces
bp, q of harmonic functions over
D and over [Copf ] (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric
system Ω = {r[mid ]k[mid ]eikϕ}k∈ℤ
is never a basis of b1, 1 and b∞, 0 for any weighted
L1-norm and L∞-norm over D.
The same result holds in the special case of Bargmann–Fock space over [Copf ] (with respect to the weighted
L1-norms and L∞-norms) which
answers a question of Garling and Wojtaszczyk. On the other hand
examples are given of weighted L1-norms and L∞-norms
over [Copf ] where Ω is indeed a basis of b1, 1 and b∞, 0.
Moreover, using similar methods, a weight is constructed on D where
b∞, ∞ is not isomorphic to l∞ which
shows that there are weighted spaces whose Banach space classifications differ completely from those which
have been characterized so far.