Mathematical Preliminaries for Transforms, Subbands, and Wavelets

Abstract

This chapter reviews some of the mathematical background necessary for the study of transforms, subbands, and wavelets. The techniques used to obtain compression involve manipulations and decompositions of (sampled) functions of time. The mathematical framework needed is provided through the concept of vector spaces. Two vectors are said to be orthogonal if their inner product is zero. A set of vectors is said to be orthogonal if each vector in the set is orthogonal to every other vector in the set. The inner product between a vector and a unit vector from an orthogonal basis set provides the coefficient corresponding to that unit vector. This chapter discusses vector space. A vector space consists of a set of elements called “vectors” that have the operations of vector addition and scalar multiplication defined on them. The results of these operations are also elements of the vector space.