Introduction to Frames and Riesz Bases - Part 1

Explore the fundamentals of frames and Riesz bases in this 54-minute mini-course lecture delivered by Ole Christensen from the Technical University of Denmark. Delve into key concepts such as orthonormal bases, Riesz sequences, and Bessel sequences. Examine the expansion property for non-bases and understand the frame decomposition. Compare general frames with tight frames and learn their characterizations. Analyze the differences between frames and Riesz bases, and investigate general dual frames. Discover the characterization of all dual frames and explore an example of Sigma-Delta quantization. Conclude by examining a frame where no subsequence forms a basis. This lecture, part of the Focus Program on Analytic Function Spaces and their Applications, provides a comprehensive introduction to these important mathematical concepts.

Intro
Goal and scope
Characterizations of ONB's
Riesz sequences
Key properties of Riesz bases = {U }
The expansion property for a non-basis
Bessel sequences
The frame decomposition
General frames versus Tight frames
Characterizations of frames
Frames versus Riesz bases
General dual frames
Characterization of all dual frames
An example: Sigma-Delta quantization
A frame where no subsequence is a basis