# Two. Mathematical Preliminary

Published Online: 24 NOV 2010

DOI: 10.1002/9780470926994.ch2

Copyright © 2011 John Wiley & Sons, Inc.

Book Title

## Fundamentals of Wavelets: Theory, Algorithms, and Applications, Second Edition

Additional Information

#### How to Cite

Goswami, J. C. and Chan, A. K. (2011) Mathematical Preliminary, in Fundamentals of Wavelets: Theory, Algorithms, and Applications, Second Edition, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9780470926994.ch2

#### Publication History

- Published Online: 24 NOV 2010
- Published Print: 10 JAN 2011

#### Book Series:

#### Book Series Editors:

- Kai Chang

#### Series Editor Information

Texas A&M University, USA

#### ISBN Information

Print ISBN: 9780470484135

Online ISBN: 9780470926994

- Summary
- Chapter
- References

### Keywords:

- biorthogonality;
- digital signals;
- linear spaces;
- linear transformation;
- matrix algebra;
- orthogonality;
- riesz basis;
- vector spaces

### Summary

The purpose of this chapter is to familiarize the reader with some of the mathematical notations and tools that are useful in an understanding of wavelet theory. For a more detailed discussion of functional spaces, the reader is referred to standard texts on real analysis. A fundamental understanding of topics in digital signal processing, such as sampling, the z - transform, linear shift - invariant systems, and discrete convolution, are necessary for a good grasp of wavelet theory. In addition, a brief discussion of linear algebra and matrix manipulations is included that is very useful in discrete - time domain analysis of filter banks. Biorthogonal representation is a possible alternative to overcoming the constraint in orthogonality and producing a good approximation to a given function. The Shannon basis is an example of a Riesz basis that is orthonormal, since the spectrum of the Shannon function is one in the interval.

#### Controlled Vocabulary Terms

digital signal processing; digital signals; information theory; matrix algebra