Approximate Schauder Frames for Banach Sequence Spaces

Series
Dissertation Defense
Time
Friday, April 16, 2021 - 4:00pm for 1.5 hours (actually 80 minutes)
Location
ONLINE
Speaker
Yam-Sung Cheng – Georgia Institute of Technology – ycheng61@gatech.edu
Organizer
Yam Sung Cheng

The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space 1. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real sequence space n1. We prove that for any Nn, FUNTFs of length N exist for real n1. To show the existence of FUNTFs, 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 n1 of lengths 2n1 and 2n2 that do not have repeated elements.

The second main topic of this thesis pertains to normalized unconditional Schauder frames for the sequence space 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 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 1-type of frame inequality.

Bluejeans link to meeting: https://bluejeans.com/544995272