Fourier Series and Finite Abelian Groups William C. Schulz Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 860. INTRODUCTION The purpose of this paper is to illustrate how Fourier Series and the Fourier Transform appear as generalizations of natural activities related to the Group Algebra of a finite Abelian Group. The connection between Fourier activities and Representation theory has long been nown and in this paper I want to illustrate this at the most elementary possible level. I will loo at the Group Algebra with the ultimate goal of setting up a Fourier Transform in the finite group situation which will then generalize to the ordinary Fourier Series and Fourier Transform when we change to other groups.. THE GROUP ALGEBRA We will illustrate the ideas mostly by example on the smallest possible Abelian group on which we can wor; C, the cyclic group of order four. The fact that C is cyclic has no effect on our examples and will be ignored; I am using C because it is the smallest group for which the representations require complex numbers. Because several inds of multiplication will occur in our discussion we will use for multiplication in the group G C. Our group G is e e e 3 e e e e e 3 e e e e 3 e e e 3 e 3 e e e e e e e e 3 and we write e e e. It is critical for our purposes that C is commutative as this means all the irreducible representations are one dimensional. For any commutative Ring F it is possible to form F(G) made up of formal lineear combinations of group elements with coefficients from F. For example, if F Z then f e + 3e e 3 g e + 5e + e 5 Aug 008
are elements of Z(G) and multiplication is f g (e + 3e e 3 ) (e + 5e + e ) e e + 0e e + e e + 6e e + 5e e + 3e e 8e 3 e 0e 3 e e 3 e e + 0e + e + 6e + 5e 3 + 3e 8e 3 0e e 7e + e + 7e 3 8e We will have several more examples of this sort but in order not to brea up the flow they are in Section 5 EXMAPLES. The reader may wish to loo at this section from time to time as we coninute our development. The examples are arranged in the same order as the development. Let s now loo more closely at multiplication in the group algebra. We can then write f f i e i g g j e j and we have then where i f g h i h e e i e je f i g j. For later purposes we will want to loo at these formulas from a different angle. Instead of seeing a group algebra element is regarded as a linear combination of group elements we can loo at is as a function from the group to the commutative ring F. Thus the element f f e +f e +f 3 e 3 +f e we regard as the function: j f(e ) f, f(e ) f, f(e 3 ) f 3, f(e ) f Rewritten in this manner the function f g has the formula (f g)(e ) e i e je f(e i )g(e j ). where h(e ) h and h is given by the formula above. 3. HOMOMORPHISMS OF THE GROUP AND THE GROUP ALGEBRA Because we want to use ordinary Group Represention theory we will now specialize to the case where F C, the complex numbers. Now we consider irreducible representations of G which, because G is commutative, are all one
dimensional. These irreducible representations are functions φ : G C which satisfy φ(x y) φ(x)φ(y). Since x e for all x G, we have φ(e) φ(x ) φ(x) for all x G. so that, in our example, φ(x) is always a fourth root of unity in C. (In general φ(x) will be a th root of unity.) It is natural to extend the homomorphism φ to the Group Algebra by simply using linearity: φ(f) φ( f i e i ) f i φ(e i ) C and we now as how φ treats multiplication in the Group Algebra. We see that, with our previous notation, φ(f g) φ( h i e i ) h i φ(e i ) ( ) f j g φ(ei ) i e j e e i f j g φ(e j e ) j, ( )( ) f j φ(e j ) g φ(e ) j φ(f)φ(g) so that φ is an algebra homomorphism φ : C(G) C. The irreducible representations of G are easily found; they are e e e 3 e φ φ i i φ 3 φ i i We notice that the columns are orthogonal; to be more precise we put an inner product on C defined by < (x, x, x 3, x ) (y, y, y 3, y ) > x i y i and with this inner product the columns are orthogonal (as are the rows). We will now put an inner product on the group algebra by simply using the inner product in C defined above. We define (f, g) f(e i )g i i 3 f i g i i i
At this point we will begin to explicitly explore the analogy between our situation with G {e, e, e 3, e } and Fourier series. For Fourier series the group is T, the one dimensional torus, or more familarly the unit circle in C. The group elements are parametrized by t R/Z so that a group element a e it. Thye group homomorphisms are φ n whereφ n (a) e int. The group algebra is the set of continuous functions f : T C; that is the continuous periodic functions with period π. The inner product is (f, g) π π 0 f(t)g(t) dt We note the changes; since the group is a continuous group the linear algebra method f i f ie i is no longer available to us, the sum is replaced by an integral, and the size of the group is replaced by its measure π. Note that in both cases if we tae the sum of the homomorphism which sends everything to the identity (φ for G and φ 0 for T ) over the group we get the measure of the group, ( for G and π for T ). A Stiljes measure would do both jobs at once. Another difference is there is a countable set of homomorphisms φ n, n...,, 0,,,... for T. This is characteristic for abelian continuous groups which are compact. We will need the orthogonality relations. Notice that the conjubage of a homormorphism is also a group homomorphism; from the table φ φ, φ φ, φ 3 φ 3, φ φ If we regard the φ i as elements of the group algebra then (φ i, φ j ) φ i (e )φ j (e ) φ i (e )φ j (e ) δ ij as we see from the table and fairly easy to prove for the irreducible representations of any Abelian Group. Note that according to the definitions we have φ i (f) φ i f (e ) f φ i (e ) f φ i (e ) Note that the f are not conjugated.. FOURIER THEORY Now we wish to introduce the Fourier Transform for a finite Abelian group. In this theory the size or measure of the group must show up somewhere. This is for our G C and it is T π for the circle group. Exactly where the size is placed is a matter of choice and I have chosen to place it so it corresponds to the position in classical Fourier series formulas.
We define the Fourier Transform for G C, with f i f ie i by F(f) ( φ (f), φ (f), φ 3 (f), φ (f) ) This will give us the set of coefficients we need to reconstruct f as a linear combination of the homomorphisms φ i. In fact, setting c i φ i(f) we will have f c i φ i φ i (f)φ i i i To see this is correct, let us apply the formula to e j : i φ i (f)φ i (e j ) f j f(e j ) ( φ i f e )φ i (e j ) i ( ) f φ i (e ) φ i (e j ) i ( ) f φ i (e )φ i (e j ) i f δ j It is also possible to write the formula for Fourier coefficients in terms of the inner product: c i φ i(f) φ i( f j e j ) φi (e j )f(e j ) (φ i, f) φi (e j )f j These formulas illustrate the algebraic heart of Fourier thoery or Harmonic analysis. In more complex situations finite sums become infinite sums or integrals, there are convergence questions which are critical, and the series may not converge for every function in the group algebra. However we will always find the basic algebraic seleton of the last two calculations. For example, we will always have the interchange of order of sums that we see in the first calculation although it may turn out to be an interchange of a sum and an integral or interchange of order in a double integral, and may require quite delicate analysis to justify. The end result will always be a representation of the element of the group algebra (except for possible convergence difficulties) in terms of sums or integrals of homomorphisms multiplied by suitable coeeficients. 5
Now let s tae a quic loo at the Fourier series analog, where G T and φ n is the homomorphism given my e int. The size of the group is found integrating the unit homomorphism over the group: T π 0 φ 0 (t) dt π 0 e i0t dt π The Fourier Transform which is this case is the series of coefficients of the Fourier series is given by F(f) (..., c, c 0, c, c,... ) where π c n e int dt 0 φ n(f) (φ n, f) just as in C but with integral instead of sum. The formula that reconstitutes f as a weighted sum of homomorphisms is f(t) c n φ n n n c n e int just as in C. Of course to really prove these formulas one must bring some analysis on stage. Next we wish to compare the inner product (f, g) with the inner product ( ˆf, ˆf). This is easy due to the orthogonality relations. We tae ˆf (c, c, c 3, c ) c i φ i(f) ĝ (d, d, d 3, d ) d i φ i(g) so we have, with the C inner product, (f, g) c i d i φ i (f) φ i (g) i ( )( ) f j φ i (e j ) g φ i (e ) i j ( ) f j g φ i (e j )φ i (e ) j f j g δ j j f j g j j (f, g) 6 i
where (f, g) is the inner product in C(G). This is the Plancherel Theorem. If we set f g we have ˆf f For G T the corresponding formula is c n π n π 0 f dt Next we want to find what happens to the Fourier Transforms when functions are multiplied in the group algebra. Recall that f g h e where h f i g j e i e je which we recall is a very natural operation in the group algebra. Now if ˆf (c, c, c 3, c ) ĝ (d, d, d 3, d ) we as what is f g? Once again tis is easy and rests on the defining property of the homomorphisms φ (e i e j ) φ (e i )φ (e j ). We have, using componentwise multiplication in C, ˆfĝ (c d, c d, c 3 d 3, c d ) To decode the meaning we calculate, with h f g, c d which means that we have φ (f)φ (g) φ ( f i e i )φ ( g j e j ) i j f i g j φ (e i )φ (e j ) ij f i g j φ (e i e j ) ij f i g j φ (e l ) e i e je l φ ( h ) l h l φ (e l ) φ (h) ˆfĝ ĥ f g 7
The Fourier series analog of this would be π The Fourier series of π f(t u)g(u)du is 0 n c nd n e int. 5. SOME EXAMPLES In this section we present examples of the theory for the group C using f e + 3e e 3 g e + 5e + e f g 7e + e + 7e 3 8e Recall that we previously have calculated f g. We first find the Fourier Transform ˆf of f and show that f reconstitutes properly as a weighted sum of homomorphisms. To find ˆf we can use either of the formulas c i (φ i, f) φ i(f) (φ, f) + 3 + ( ) (φ, f) + ( i) 3 + ( ) ( ) 6 3i (φ 3, f) + ( ) 3 + ( ) 5 (φ, f) + i 3 + ( ) ( ) 6 + 3i and so ˆf ( ) (, 6 3i, 5, 6 + 3i, 6 3i, 5, 6 + 3i ) Our theory tells us that f i c i φ i φ + 6 3i φ + 5 φ 3 + 6 + 3i To chec the correctness we will evaluate both sides on e. f(e ) φ (e ) + 6 3i φ (e ) + 5 φ 3(e ) + 6 + 3i φ (e ) 6 3i () + i + 5 6 + 3i ( ) + ( i) ( ) + 3 + 6i + 5 + 3 6i 3 which is correct. In a similar way we find ĝ (, i,, + i ) g φ + i 8 φ φ 3 + + i φ φ
Now we will illustrate the Plancherel theorem: From our examples we have ˆf f ( ˆf, ĝ) (f, g) f + 3 + ( ) 9 ˆf ( + 6 3i + ( 5) + 6 + 3i ) 6 ( ) + 5 + 5 + 5 6 6 6 9 f (f, g) + 5 + 0 + 0 9 ( ˆf, ĝ) + 6 + 3i i + 5 6 3i ( ) + + i ( ) + 9i + 0 + + 9i 8 8 38 9 Finally we illustrate the formula ˆfĝ f g We have f g 7e + e + 7e 3 8e and, for example, φ (f g) φ (f g) 7 i 7 8i 30i Doing the other three calculations we have f g ( ) 8, 30i, 0, 30i (, 5 5 i, 5, i ) f g (, 5 8 i, 5, 5 8 i ) ˆfĝ (, 6 3i ( 6 3i,, 5, 6 + 3i i, 5 (, 5 8 i, 5, 5 8 i ) )( i, i,, + i ), 6 + 3i + i 6. THE CLASSICAL FOURIER TRANSFORM For the classical Fourier transform the formulas are ˆf(s) π e ist f(t)dt f(t) 9 e ist ˆf(s)ds )
which bear close resemblence to the Fourier series case but the infinite sum has been replaced by an integral in the second formula. These formulas may be derived heuristically by applying the series formula to functions periodic with period l, letting l, and regarding the Fourier series as a Riemann sum for the integral in the second formula. This procedure would be rather difficult to justify, but once the formulas are nown it is possible to prove them by nearly the same methods as we use to prove the convergence of Fourier series. The similarities and differences to the situation for finite groups and Fourier series are interesting. The group for Fourier transforms is R with addition as the operation. For the group algebra one can use functions f(t) with f(t) dt finite, although it is more productive the use f(t) dt finite, though this complicates the definition of ˆf Again f(t). Again f(t) is a weighted sum of homomorphisms although in this case the sum manifests as an integral. This difference of sum versus integral is related to the compactness of C and T versus the non-compactness of R. The homomorphisms of R are again functions of the form e ist but here s is a real number rather than an integer as before. Note that the non-compactness of R means that we no longer have to wor with. Thus we cannot directly use the formulas we derived in the finite group case to guess at the formulas for the Fourier transform, although as I indicated above they can be Fourier transform formulas can be derived with a little analytic tricery. In spite of this, the formulas we derived for the Plancherel theorem and for f g remain valid after is replaced by π. This hints at a more general theory (for locally compact Abelian groups) which of course exists. 0