7.8 Functions of several variables

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Remark 7.8.1. Fourier expansions of functions from [−_, _] are simple

enough conceptually and easy to compute given current machines. This

is the start of the Fourier Transform which has been used extensively for

the analysis of signals in Engineering, particularly Electrical Engineering.

But these days we are often confronted with 2-dimensional signals. They

are sometimes called images. So two dimensional transforms are important.

Also, it will turn out to be helpful to use Fourier Theory in finding solutions

to the partial differential equations which arise in the study of diffusion and

wave propagation. These are frequently defined on three dimensional regions

of space, so it will be necessary to know how to do an analysis of functions

from rectangles and cubes.

Fortunately this is very easy.

Definition 7.8.1. Let Q _ R2 denote the square [−_, _] × [−_, _]

Definition 7.8.2. f : Q −! R is rectangularly piecewise continuous iff

there is a decomposition of Q into rectangles which overlap only on their

boundaries, such that f is continuous on the interiors of the rectangles, and

bounded on their boundaries.

Remark 7.8.2. This is more restrictive than necessary but makes life easier.

Definition 7.8.3. RPC(Q) denotes the set of equivalence classes of rectangularly

piewise continuous functions from Q, where two functions are equivalent

iff they differ only on a set of area zero.

154 CHAPTER 7. FOURIER THEORY

Definition 7.8.4. For all f, g 2 RPC(Q),

< f, g >=

Z

Q

f(s, t)g(s, t)

is the standard inner product.

Remark 7.8.3. Calling it an inner product doesn’t make it one:

Exercise 7.8.1. Prove this is an inner product on RPC(Q)

Proposition 7.8.1. The space RPC(Q) has the functions

{1, cos(nx) cos(my), cos(nx) sin(my),

sin(nx) cos(my), sin(nx) sin(my) : n,m 2 Z+      

as an orthogonal basis.

Proof:

The orthogonality is very simple: For example

Z

Q

cos(nx) cos(my) cos(px) sin(qy)

is clearly zero since it is

Z _

−_

cos(nx) cos(px)

Z _

−_

cos(my) sin(qy)

and both terms in the product are zero.

The proof of convergence goes the same way as for one dimension. We need a

Weierstrass theorem for functions of two variables that says we can approximate

uniformly any continuous function on Q by trigonometric polynomials,

and we also need to prove that a piewise continuous function can be approximated

in the L2 metric by a continuous one; you can try to draw the

picture for a discontinuity between rectangular regions to show we can do

this. Take my word for it that there is indeed a Weierstrass theorem for

squares, and verify that we can join up two discontinous patches either side

of a line discontinuity. _

Figure 7.8 shows the problem of patching over a fault line, and although it

is a little more complicated it can be done.

This means that it is possible to do approximations to functions of two

variables on any rectangular region. By composing with suitable functions

7.8. FUNCTIONS OF SEVERAL VARIABLES 155

Figure 7.8: A fault line discontinuity

from other regions to rectangles (embeddings almost everywhere) we can get

Fourier series on these regions too. For example, discs, using the inverse of

the polar coordinate transformation.

Such series are called double Fourier series.

And it is a small step to doing it for solid regions of R3 with triple Fourier

series. Again, if f : Q −! R is piecewise smooth, then the Fourier sequence

converges to f pointwise, (not just in the L2 metric) in both the two and

three dimensional cases.

This is just the start of a big area of Mathematics. The generalisations

include doing it for functions defined on all R (The Fourier Transform) and

using other orthogonal bases besides the Trigonometric functions (generalised

Fourier theory). It is not a coincidence that the sine and cosine functions are

the solutions to the ordinary differential equation

¨x = −x

In fact their orthogonality comes about precisely because they are eigenvectors

of a linear operator on an infinite dimensional space. This leads to

considering other more complicated linear operators each with their own family

of orthogonal functions, although in general we have to take a somewhat

different inner product. Check out the Bessel functions in Mathematica.

The Matlab toolkit for doing image processing can be explored by those

having access to it.

156 CHAPTER 7. FOURIER THEORY

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