Lecture 1 Introduction to Fourier Analysis Jan 7 , 2005 Lecturer : Nati Linial

Consider a vector space V (which can be of finite dimension). From linear algebra we know that at least in the finite-dimension case V has a basis. Moreover, there are more than one basis and in general different bases are the same. However, in some cases when the vector space has some additional structure, some basis might be preferable over others. To give a more concrete example consider the vector space V = {f : X → R or C} where X is some universe. If X = {1, · · · , n} then one can see that V is indeed the space Rn or Cn respectively in which case we have no reason to prefer any particular basis. However, if X is an abelian group1 there may be a reason to prefer a basis B over others. As an example, let us consider X = Z/nZ, V = {(y0, · · · , yn)|yi ∈ R} = Rn. We now give some scenarios (mostly inspired by the engineering applications of Fourier Transforms) where we want some properties for B aka our “wish list”: An abelian group is given by 〈S, +〉where S is the set of elements which is closed under the commutative operation +. Further there exists an identity element 0 and every element in S has an inverse.