Two. Mathematical Preliminary

  1. Jaideva C. Goswami and
  2. Andrew K. Chan

Published Online: 24 NOV 2010

DOI: 10.1002/9780470926994.ch2

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

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

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

  1. Published Online: 24 NOV 2010
  2. Published Print: 10 JAN 2011

Book Series:

  1. Wiley Series in Microwave and Optical Engineering

Book Series Editors:

  1. Kai Chang

Series Editor Information

  1. Texas A&M University, USA

ISBN Information

Print ISBN: 9780470484135

Online ISBN: 9780470926994

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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