2011/08/31 by Diana T. Stoeva, Stoeva, Diana T.
Mathematics · #40A05 #42C15 #47A05 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #math.FA #msc:40A05 #msc:42C15 #msc:47A05
paper · pdf · doi:10.48550/arxiv.1108.6282
The paper was extended
openalex publication_date 2011/08/31 · openalex created_date 2016/06/24 · arxiv created 2016/09/09 · arxiv updated 2016/09/12 · openalex updated_date 2026/07/28
In this paper we consider series expansions via a frame and a non-frame and the possibilities for interchange of the two sequences, both in the Hilbert and Banach space setting. First we give a characterization of frame-related concepts in Banach spaces (atomic decompositions, Banach frames, Xd-Riesz bases, Xd-frames, Xd-Bessel sequences, and sequences satisfying the lower Xd-frame condition). We also determine necessary and sufficient conditions for operators to preserve the type of the concepts listed above. Then we discuss differences and relationships between expansions in a Banach space and its dual space when interchanging the involved sequences. Finally, we apply some of the results to answer problems in Hilbert frame theory. We show that interchanging a frame and a non-Bessel sequence in series expansions is not always possible (leading to differentiation of analysis- and synthesis-pseudo-duals of a frame) and determine an appropriate subspace where interchange can be done. We characterize all the synthesis-pseudo-duals of a frame and determine a class of frames whose synthesis-pseudo-duals (resp. analysis-pseudo-duals) are necessarily frames. We also investigate connections between the lower frame condition and series expansions. Examples are given to illustrate statements in the paper and to show the optimality of some results.