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The Fourier series of the elements in the generalized Bergman spaces b p , 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 ] e ik ϕ } k ∈ℤ is never a basis of b 1, 1 and b ∞, 0 for any weighted L 1 -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 L 1 -norms and L ∞ -norms) which answers a question of Garling and Wojtaszczyk. On the other hand examples are given of weighted L 1 -norms and L ∞ -norms over [Copf ] where Ω is indeed a basis of b 1, 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.

Weighted spaces of harmonic and holomorphic functions on the unit disc are studied. We show that for all radial weights which are not decreasing too fast the space of harmonic functions is isomorphic to c0. For the weights that we consider we completely characterize those spaces of holomorphic functions which are isomorphic to c0. Moreover, we determine when the Riesz projection, mapping the weighted space of harmonic functions onto the corresponding space of holomorphic functions, is bounded.

Weighted spaces of harmonic and holomorphic functions on the unit disc are discussed. It is shown that these spaces are always subspaces of c0. Moreover, for many weights, it is shown that the weighted space of holomorphic functions has a basis.