On the Fourier Series of Unbounded Harmonic Functions

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.