By V. P. Havin, N. K. Nikol’skii
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Additional info for Complex Analysis and Spectral Theory: Seminar, Leningrad 1979/80
O,n r,otati, On-invariant subs,paces of ,~, ~ ,0< P < ~' Let G group. Let be a compact abelian group. Denote by P A be a subset of ~ the dual . Recall that ~A ~ ~ ) 50 denotes the closure (in LP (~) ) of the linear span of the set A ( 0 ~ p ~ ÷ o 0 ) . It is clear that ~ A Q G ) is a closed rotation-invariant subspace of ~ ( G ) ( 0 < p < ~ 0~) . It is well-known that these subspaces are the only invariant l) subspaces of ~ ( G ) when P ~ [ i ~ ~ ) . e. g. 0 C * - ~ ~o~'~(~) , where[6~}~ A is a summable family of non-zero numbers.
8 fall for non-locally holomorphlc spaces. §3. 2 that (_A~+) scalar of all does n o t depend on ~ . 47 (T ~) Le~ B P ~'1]) ~-~-C d e n o t e t h e s e t Of a l l holomorphlc functions such that ~f t~(~)t <+CO . 7 and the induction prove the following in , it is not hard to • Then THEOREM 3 . 2 . ,~-~ ~ ( A I / P ' ~ ct r uch that f ~-~- for ~ ~ ~ ~+~ (~r ~ ) . ,o~ ~ t i c ~ H[(T'5 6Ui+ )~¢ deZi~es a linear oonti=uo~ functional on by t h e f o r m u l a (3)• I' 1 ~ o o ~ . Let ~ e ( A ~ " ) . Chec~ that the f o = ~ (3) ~,fines a linear continuous functlon-] on H~ ( T ~) • Note t h a t Hence 48 It remains to note that 1, ~' ~-~r__~a~-~)'1 fo .
Et ~' X : a all % > 0 , • Then be ~ l o c a l l ~ hole~orphic space, O