Towards characterizations of approximate Schauder frame and its duals for Banach spaces

Lipschitz p-Approximate Schauder Frames

“…Holub characterized frames for Hilbert spaces using standard orthonormal basis for the standard Hilbert space [20]. This result has been derived for Banach spaces in [21]. We show that such a result can be derived for Lipschitz p-ASFs.…”

“…In 1995, Li derived a characterization of dual frames using standard orthonormal basis for ℓ 2 (N) [23]. For Banach spaces, such a characterization using canonical Schauder basis for ℓ p (N) is derived in [21]. Now we derive such characterization for Lipschitz p-ASF.…”

“…is a well-defined invertible bi-Lipschitz map and Whenever M = X, and f n 's are all linear, Definition 2.1 reduces to definition of p-ASF given in [21]. It is important to note that the partial sums of series in (iii) of Definition 2.1 need not be inside M (which may not be as it is only a subset) but only demanding limit has to be inside M. Definition 2.1 says that there are a, b, c, d > 0 satisfying following:…”

“…In 2014, Thomas, Freeman, Odell, Schlumprecht and Zsak [16,27,28] introduced the notion of approximate Schauder frames as a generalization of notion of Schauder frames by Casazza, Dilworth, Odell, Schlumprecht and Zsak [4] (also see [10]). In 2021, Krishna and Johnson characterized some classes of approximate Schauder frames [21]. In 2022, Krishna and Johnson introduced metric frames which have surprising connections with subsets of Banach spaces using Lipschitz-free Banach spaces [22].…”

Unexpected Uncertainty Principle for Disc Banach Spaces

“…Banach space frame theory which is modeled on classical Lebesgue sequence spaces [10] and the theory of unbounded frames for Hilbert and Banach spaces [11][12][13][14][15][16][17] naturally gives the following definition. Definition 2.2.…”