ABSTRACT
In elementary statistics, observations are numbers (scalars), and are elements of the real line. The first step in analysis of functional data is to convert functions observed at discrete time points to functional objects using basis expansions. For example, if the basis consists of trigonometric functions, each functional object is a function defined on the whole interval because each sine and cosine function is defined on the whole interval. An excellent and very extensive account of mathematical foundations of Functional Data Analysis (FDA) is given in the monograph of Hsing and Eubank. Linear transformation play a very important role in statistics. In FDA, many types of regressions are considered; they involve linear transformations of functions to functions, functions to scalars and vectors to functions. This chapter introduces some basic concepts related to general linear transformations to help the reader follow the chapters on functional regressions.
