Harry F. Davis - Fourier Series and Orthogonal Functions

This appendix is intended exclusively for readers having some prior knowledge of modern algebra, including group theory and the theory of rings and ideals. Most of the necessary background material can be found in Birkhoff and MacLane, A Survey of Modern Algebra (Macmillan, 1953), referred to as [BM] in the following discussion.

Let G be any finite group, with elements 1, 2, , n . If the group operation is written in multiplicative form, for any i, j we have i j = k for some k. We tentatively define the group algebra of G in the following manner. The elements of the group algebra are formal linear combinations c 11 + c 22 + ... + c n n , where the cj s are complex numbers ( j = 1, 2, , n ). The product of two elements of the group algebra is defined by formally multiplying them and simplifying using the group operation. That is,

(1)

(In [BM, page 240], a more general definition is given, which permits the scalars to be in an arbitrary field; most of the following discussion is not valid, however, unless the scalars are complex numbers.)

Since i j = k for some k, we can write the right side of (1) in n the form where z k is the sum of those products x i y j for which i j = k . Our first object is to show that, when rewritten in a suitable form, the product in (1) is essentially the convolution product of two functions.

Let f be a complex-valued function on G with values given by f ( j ) = x j . Similarly, let g be defined by g ( j ) = y j . Then we can write (1) in the form

(2)

where h ( k ) is the following sum

(3)

taken over those pairs i and j for which i j = k . Since i = k j 1 this can be written

(4)

This function h is called the convolution product of the functions f and g, in that order, and is denoted f * g.

It is easy to see that (4) can also be written in the form

(5)

We see from this discussion that there is a natural one-to-one correspondence between elements of the group algebra x + x + ... + x n n and functions f ( j ) = x j ( j = 1,2, ...; n ) , whereby the product of two elements corresponds to the convolution product of the corresponding functions. (This correspondence is also a linear isomorphism.) It is more convenient for our purposes to work with functions, so we shall now drop the tentative definition previously given for the group algebra, and use instead the definition given in the following paragraph. Also, we shall consider only commutative groups and, since it is more suggestive of Fourier series, we shall change to additive notation , so that (4) can be written in the form

(6)

At this point the reader should ignore all of the preceding material in this section, which was intended to serve simply as an introduction for the following.

Let G be any finite commutative group, with the group operation written in additive form. [Thus, we write y instead of y 1 and x y instead of xy 1.] Let L ( G ) denote the collection of all complex-valued functions with domain G. The sum of two functions and the product of a function by a complex scalar are defined in the usual way, so that L ( G ) is an n-dimensional complex linear space where n is the order (i.e., number of elements) of G . Define a multiplication f * g in L ( G ) by (6), where the sum is over all elements y in G . This multiplication can be shown to be associative, i.e., ( f * g ) * p = f * ( g * p ) whenever f, g , and p are elements of L ( G ). Indeed, L ( G ) satisfies all the axioms of a linear associative algebra [BM, page 239] over the complex field. We call L ( G ) the group algebra of G.

The group algebra L ( G ) contains a unity element , which is the function defined by (0) = 1, ( x ) = 0 if x 0. (Here we are using 0 as the identity element of G , as well as the number zero.) That * f = f * = f for every f in L ( G ) is easily shown by substituting into (6).

Since G is commutative (the term Abelian is used instead of commutative in many books), it is easy to show that f * g = g * f for every pair of elements f and g in L ( G ).

EXAMPLE: Let G be the additive group of integers modulo 4 [BM, pages 27 and 131]. The group operation table is

Notice that 3 + 2 = 1, by this addition table, and therefore 1 2 = 3; similarly, 1 3 = 2.

Now suppose that f and g are defined as shown in the following table. It will follow that f * g is the function shown in the last line of the table.

For example, if we take x = 1 in (6) we obtain

Returning to the general theory, a subspace L ( G ) is called a subalgebra if f * g is in the subspace whenever f and g are in the subspace. For example, the collection of all functions f with the property f (2) = f (3) = f (4) = 0 is a one-dimensional subalgebra, as the interested reader can verify for himself.