Approximate Schauder Frames for Banach Sequence Spaces

Abstract

The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space l_1^n. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real sequence space l_1^n. We prove the existence of FUNTFs for real l_1^n. To do so, specific examples are constructed for various lengths. These constructions involve repetitions of frame elements. However, a different method of frame constructions allows us to prove the existence of FUNTFs for real l_1^n of lengths 2n−1 and 2n−2 that do not have repeated elements.

The second main topic of this thesis pertains to normalized unconditional Schauder frames for the sequence space l_1. A Schauder frame provides a reconstruction formula for elements in the space, but need not be associated with a frame inequality. Our main theorem on this topic establishes a set of conditions under which an l_1-type of frame inequality is applicable towards unconditional Schauder frames. A primary motivation for choosing this set of hypotheses involves appropriate modifications of the Rademacher system, a version of which we prove to be an unconditional Schauder frame that does not satisfy an l_1-type of frame inequality.