We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 23 February 2017
Introduction
In the previous chapters, we have developed the general ideas necessary to be able to speak with precision about wavelets and wavelet series. The key concepts were those of Hilbert spaces and of orthonormal bases and Fourier series in Hilbert spaces, and we have seen that Haar wavelet series are
examples of such series.
But although we have developed a substantial body of relevant background material, we have so far developed no general theory of wavelets. Indeed, the simple and rather primitive Haar wavelet is the only wavelet we have encountered, and we have not even hinted at ways of constructing other wavelets, perhaps with ‘better’ properties than the Haar wavelet, or at the desirability of being able to do so. These are the issues that we turn to in the final chapters of the book.
Specifically, our major remaining goals are as follows:
• to develop a general framework, called a multiresolution analysis, for the construction of wavelets;
• to develop within this framework as much of the general theory of wavelets as our available methods allow; and
• to use that theory to show how other wavelets can be constructed with certain specified properties, culminating in the construction of the infinite family of wavelets known as the Daubechies wavelets, of which the Haar wavelet is just the first member.
The main aim of the present short chapter is to introduce the idea of a multiresolution analysis by defining and studying the multiresolution analysis that corresponds to the Haar wavelet.
The Haar Wavelet Multiresolution Analysis
The concept of a multiresolution analysis was developed in about 1986 by the French mathematicians Stéphane Mallat and Yves Meyer, providing a framework for the systematic creation of wavelets, a role which it still retains. Mirroring all of our discussion of wavelets up to this point, we will introduce the idea of a multiresolution analysis by first examining it carefully in the specific case of the Haar wavelet. As we have found in our previous discussion, the Haar wavelet is simple enough for us to see in a very concrete fashion how the various constructions and arguments work.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
