INTRODUCTION

We start this chapter with an intuitive derivation of the main result for Fourier integrals from the fundamental theorem of Fourier series. A mathematical rigorous treatment of the results obtained is postponed until chapter 7. In the present chapter the Fourier integral will thus play a minor role. First we will concentrate ourselves on the Fourier transform of a non-periodic function, which will be introduced in section 6.2, motivated by our intuitive derivation. After discussing the existence of the Fourier transform, a number of frequently used and often recurring examples are treated in section 6.3. In section 6.4 we prove some basic properties of Fourier transforms. Subsequently, the concept of a ‘rapidly decreasing function’ is discussed in section 6.5; in fact this is a preparation for the distribution theory of chapters 8 and 9. The chapter closes with the treatment of convolution and the convolution theorem for non-periodic functions.

LEARNING OBJECTIVES

After studying this chapter it is expected that you

1. - know the definition of the Fourier transform

2. - can calculate elementary Fourier transforms

3. - know and can apply the properties of the Fourier transform

4. - know the concept of rapidly decreasing function

5. - know the definition of convolution and know and can apply the convolution theorem.

An intuitive derivation

In the introduction we already mentioned that in order to make the basic formulas of the Fourier analysis of non-periodic function plausible, we use the theory of Fourier series.