ON THE FOURIER SERIES OF UNBOUNDED HARMONIC FUNCTIONS

Abstract

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^ikϕ}_k∈ℤ is never a basis of b_{1, 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 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.