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In this paper we consider series expansions via a frame and a non-frame andthe possibilities for interchange of the two sequences, both in the Hilbert andBanach space setting. First we give a characterization of frame-relatedconcepts in Banach spaces (atomic decompositions, Banach frames, $X_d$-Rieszbases, $X_d$-frames, $X_d$-Bessel sequences, and sequences satisfying the lower$X_d$-frame condition). We also determine necessary and sufficient conditionsfor operators to preserve the type of the concepts listed above. Then wediscuss differences and relationships between expansions in a Banach space andits dual space when interchanging the involved sequences. Finally, we applysome of the results to answer problems in Hilbert frame theory. We show thatinterchanging a frame and a non-Bessel sequence in series expansions is notalways possible (leading to differentiation of analysis- andsynthesis-pseudo-duals of a frame) and determine an appropriate subspace whereinterchange can be done. We characterize all the synthesis-pseudo-duals of aframe and determine a class of frames whose synthesis-pseudo-duals (resp.analysis-pseudo-duals) are necessarily frames. We also investigate connectionsbetween the lower frame condition and series expansions. Examples are given toillustrate statements in the paper and to show the optimality of some results.

Characterization of atomic decompositions, Banach frames, Xd-frames, duals and synthesis-pseudo-duals, with application to Hilbert frame theory