5.2.1 Fourier Transformation

The result in 3.2.2 is summarized that

forms a complete ortho-normal system in L ²(0, 2 ?), or in L ²( ??, ?).

Vector spaces treated so far are over R. In this paragraph we make use of the complex variable. Actually, the function space L ²( ??, ?) is regarded as the vector space over C if each element takes the complex value. Then, it forms the Hilbert space over C through the inner product

Using Euler's convention that

for ? ? R, we have

This implies that

forms a complete ortho-normal system in (complex) L ²( ??, ?), and the Fourier series of f ? L ²( ??, ?) is written as

Then, taking x ? = Nx, we see that each f ? L ²( ? N ?, N ?) is expanded as

with

In terms of , those relations are expressed as

Then, making N ? ? formally, we get the relation that

and

The right-hand sides of (5.20) and (5.21) are called the Fourier transformation of f( x) and the inverse Fourier transformation of , and denoted by and , respectively. Then, equality (5.21) is referred to as the Planeherel's inversion formula. Justification of those relations are done in several categories.