Gentle Introduction to Wavelet

# Gentle Introduction to Wavelet - A Gentle Introduction to...

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A Gentle Introduction to Wavelets E. Rehmi Post University of Massachusetts, Amherst Abstract: After a brief review of the fundamentals of Hilbert space theory, the rudiments of wavelet analysis are illustrated by the application of a very simple family of compactly supported, orthonormal functions known as the Haar basis. This family is generated directly from a ``mother'' function, and is shown to constitute an orthonormal basis of . Introduction The wavelet transform is a tool for carving up functions, operators, or data into components of different frequency, allowing one to study each component separately. The term wavelet was itself coined in 1982, according to [Daubechies] . Wavelet analysis may be thought of as a generalization of analysis by the Hilbert space method, wherein one forms an orthogonal basis of the space of interest. Equations in that space may then be solved in terms of the basis. Hilbert space techniques are especially useful in the solution of linear ordinary differential equations (ODEs), and permit one to reduce certain partial differential equations (PDEs) to two or more ODEs related by variables of separation . Review of Hilbert space analysis To understand the wavelet transform, it helps to have an appreciation of the basic programme of Hilbert space analysis, which is as follows: 1. Identify the inner product space of interest. An inner product space (IPS) consists of a (closed) vector space and an inner product defined on that space. For example, let V be the space of real-valued functions with domain the real line ( ), and let the inner product < f, g > be defined for all f, g in V as Also, f and g are said to be orthogonal when <f,g> = 0 . 2. Define the Hilbert space of interest.

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The norm of a function f is given in terms of the inner product as . Given the inner product from above, the norm is directly analogous to a length in Euclidean space. Using the norm, a Hilbert space is defined to be It is very natural to let p=2 , in which case the space of interest is said to contain all f that are square-integrable on V . In the ensuing discussion, we concentrate on the Hilbert space . 3.
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Gentle Introduction to Wavelet - A Gentle Introduction to...

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