2000/04/01 by Wolfgang Lusky · 3 citations
Mathematics · Psychology · #Holomorphic and Operator Theory #Algebraic and Geometric Analysis #Health, Work, and Social Studies in Poland
paper · doi:10.1112/s0024610799008443
The Fourier series of the elements in the generalized Bergman spaces bp, q of harmonic functions over D and over C (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric system Ω = r∣k∣eikφk∈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 C (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 C 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.