## Some types of convergence related to the reconstruction property in Banach spaces.(English)Zbl 1311.42082

Summary: P. G. Casazza and O. Christensen [Can. Math. Bull. 51, No. 3, 348–358 (2008; Zbl 1268.42053)] introduced and studied the reconstruction property in Banach spaces. In this paper, we discuss different types of convergence of series related to the reconstruction property in Banach space. First we discuss the uniform convergence of series associated with the reconstruction property in Banach spaces. Necessary and sufficient conditions for the uniform convergence of certain series related to the reconstruction property in Banach spaces are given. A sufficient condition for a Banach space to be finite dimensional in terms of the uniform convergence of a series related to the reconstruction property in Banach spaces is obtained. Motivated by a series of papers by Casazza, we discuss unconditional convergence of series associated with the reconstruction property in Banach spaces. A necessary condition in this direction is given. An absolute type reconstruction property in Banach spaces is also discussed which depends on the absolute convergence of series related to the reconstruction property in Banach spaces.

### MSC:

 42C15 General harmonic expansions, frames 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Zbl 1268.42053
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### References:

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