Applied Fourier Analysis by Tim Olson

Author:Tim Olson
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

4.5 Partition of Unity

The Fourier Transform gives us a very powerful tool to analyze functions. In addition, the Fourier Transform gives us a very powerful tool which allows us to analyze, design, and understand other orthonormal systems.

The question now is “What does the Fourier Transform have to say about another arbitrary orthonormal basis?” The answer is

Theorem 4.5.1

(Partition of Unity) Suppose that is an orthonormal basis for . Then

Proof:

The proof is very straight forward, as are many Fourier Analysis proofs. This emphasizes the value of Fourier Analysis as a tool.

We will simply analyze the sum in two different ways. Suppose that is an arbitrary orthonormal sequence on . Then by the Fourier Isometry,

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.