Introduction

In this chapter we study Fourier series. We use Fourier series to represent or approximate functions defined on a finite interval. In this sense Fourier series are similar to polynomials or power series. However, Fourier series are in other ways both better and more general. Fourier series are one example of a closed infinite orthonormal system in an inner product space. They are an application of the general theory presented in the previous chapter. Fourier series also have various specific properties of their own and we shall study some of them. Fourier series were first defined, not too surprisingly, by Jean Baptiste Joseph Fourier (1768-1830) about 200 years ago. That they are an “old” topic does not detract from their importance. Fourier was a mathematician and an engineer who developed these series in order to solve certain problems in partial differential equations. In the last section of this chapter, we present one application of this kind. (Fourier was a participant in the French Revolution. He was with Napoleon in the Egyptian campaign of 1798 and was considered one of the “savants” who accompanied Napoleon in this campaign. He was, for a time, governor of lower Egypt, and later Prefect of Is&re (at Grenoble).)

Introduction

The main topics to be studied in this chapter are orthogonal and orthonormal systems in a vector space with inner product, as well as various related concepts. These topics are sometimes, but not always, discussed in a basic course in linear algebra. Of central importance is the subject of infinite orthonormal systems which we present at the end of this chapter. These results will be applied in the next chapter on Fourier series. The first four sections of this chapter are a condensed review of some concepts and basic ideas (with proofs) from linear algebra. We use these facts in developing the different topics of this book. The reader will hopefully find in these sections a helpful synopsis

and review of his knowledge of the area.

Linear and Inner Product Spaces

The basic algebraic structure which we use is the linear space(often called vector space) over a field of scalars. Our “field of scalars” will always be either the real numbers R or the complex numbers <D. Elements of a linear space are called vectors. Formally, a non-empty set Vis called a linear space over a field Fif it satisfies the ollowing conditions:

1. Vector Addition: There exists an operation, generally denoted by"+", such that for any two vectors u, veV﹜ the “sum” u + v is also a vector in V.

The aim of this book is to provide the reader with a basic understanding of Fourier series, Fourier transforms, and Laplace transforms. Fourier series (and power series) are important examples of useful series of functions. Applications of Fourier series may be found in many diverse theoretical and applied areas. The same holds for integral transforms. The Fourier and Laplace transforms are the best known of these transforms and are prototypes of the general integral transforms.

Fourier series and integral transforms are theoretically based on a natural amalgamation of concepts from both linear algebra and integral and differential calculus. In other words, they are a mix of algebra and analysis. We assume that the reader is well versed in the basics of these two areas. Nevertheless, in Chapter 1 is found a somewhat concise review of some of the relevant concepts and facts from linear algebra.

The best, most efficient, and perhaps only way to learn mathematics is to study and review the material and to solve exercises. At the end of almost every section of this book may be found a collection of exercises. A set of review exercises is to be found at the end of each chapter. To truly and properly understand the subject matter, it is essential to solve exercises.

A Gaussian Hilbert space is a (complete) linear space of random variables with (centred) Gaussian distributions. This simple notion combines probability theory and Hilbert space theory into a rich and powerful structure, and Gaussian Hilbert spaces and connected notions such as the Wiener chaos decomposition and Wick products appear in several areas of probability theory and its applications, for example in stochastic processes and fields, stochastic integration, quantum field theory and limit theory for various statistics. There are also applications to non-probabilistic analysis, for example Banach space geometry and partial differential equations.

Although there are many references dealing with such applications where Gaussian spaces are treated and used, see for example Hida and Hitsuda (1976), Hida, Kuo, Potthoff and Streit (1993), Holden, Øksendal, Ubøe and Zhang (1996), Ibragimov and Rozanov (1970), Kahane (1985), Kuo (1996), Major (1981), Malliavin (1993, 1997), Meyer (1993), Neveu (1968), Nualart (1995, 1997+), Obata (1994), Pisier (1989), Simon (1974, 1979a), Watanabe (1984), there seems to be a shortage of works dealing with the basic properties of Gaussian spaces in general, without connecting them to a particular application. (One exception is the paper by Dobrushin and Minlos (1977).) This book is an attempt to fill the gap by providing a collection of the most important definitions and results for general Gaussian spaces, together with some applications to special Gaussian spaces.

In this chapter we use the theory of Gaussian Hilbert spaces to obtain the asymptotic distributions of some important random variables. In the first section, we study U-statistics. In the second section we extend the results to asymmetric statistics. In the third section we extend the results further; as a special case we obtain results for random graphs.

Note that the original variables are defined without any reference to normal variables or Gaussian Hilbert spaces; the Gaussian Hilbert space is introduced as a convenient tool to treat the asymptotic distribution.

A common theme in these results is that ‘typically’ the asymptotic distribution is normal, but in some ‘degenerate’ cases other limits occur; these other limits can be represented as variables in a Wiener chaos H:n: for some Gaussian Hilbert space H, and can for example be expressed as multiple Gaussian stochastic integrals. The explanation for this phenomenon that emerges from the proofs below is that the variable in question may be expanded as a sum, where each term converges in distribution to some chaos, and the first term is asymptotically normal. Typically, the first term dominates the sum and all others are asymptotically negligible, but in degenerate cases the first term vanishes and the sum is dominated by one or several later terms, which may converge to a higher order chaos.