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On the Fourier Series of Unbounded Harmonic Functions
Author(s) -
Lusky Wolfgang
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008443
Subject(s) - mathematics , fourier series , holomorphic function , norm (philosophy) , pure mathematics , banach space , trigonometric functions , basis (linear algebra) , trigonometric series , fourier transform , space (punctuation) , trigonometry , trigonometric polynomial , mathematical analysis , geometry , linguistics , philosophy , political science , law
The Fourier series of the elements in the generalized Bergman spaces b p , 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 ∣ 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 C (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 C 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.