Character analysis using Fourier series

Although it may seem somewhat ambitious, character analysis using Fourier series is possible, at least when a mathematical character is considered, namely a homomorphism from a given group into the multiplicative group S 1 of complex numbers whose modulus is one.

To understand how, consider the well-known and useful results in (abstract) harmonic analysis saying that the character group (the group of continuous characters) of the (additive) flat torus T = R/(2πZ) is the group of integers Z, and the character group of the (additive) real line R is R. Equivalently, any 2π-periodic continuous function f : R → S 1 satisfying f (x + y) = f (x)f (y) for each x, y ∈ R must have the form f (x) = e ik•x for some k ∈ Z, and any continuous homomorphism f : R → S 1 must have the form f (x) = e iα•x for some α ∈ R.

Familiar proofs of these results [ [8, pp. 231, 237] are usually restricted to the case of continuous mappings (although the arguments in [3,5,7,8] can be generalized), and, interestingly, they involve arguments which are somewhat not in the spirit of harmonic analysis. A natural tool in harmonic analysis is Fourier series. Using this tool, we can prove the following theorem:

x is a 2π-periodic measurable homomorphism for each k ∈ Z. On the other hand, let f : R → S 1 be a 2π-periodic measurable homomorphism. Given any k ∈ Z, the bounded function f (x)e -ik•x is 2π-periodic and integrable. Hence its integral on I = [0, 2π] is invariant under translations, i.e., I f (x + y)e -ik•(x+y) dx = I f (x)e -ik•x dx for each y ∈ R. Since f is a homomorphism it satisfies f (x + y) = f (x)f (y) for each x, y ∈ R. Therefore its kth Fourier coefficient f (k) satisfies (for each y in R)

If f (k) = 0 for each k ∈ Z, then f (x) = 0 for almost every x by the uniqueness of the Fourier expansion, contradicting the fact that |f (x)| = 1 for each x. Thus f (k) = 0 for some k ∈ Z, and then f (y) = e ik•y for all y ∈ R, as claimed.

(b) It is evident that f (x) = e iα•x is a measurable homomorphism for each α ∈ R.

On the other hand, let f : R → S 1 be a given measurable homomorphism. Since |f (2π)| = 1 there exists β ∈ [0, 1) satisfying e i2πβ = f (2π). Define g : R → S 1 by g(x) = e iβ•x . Let h = f /g. Then h is well defined and it is a quotient of measurable homomorphisms. Hence h is a measurable homomorphism by itself.

In addition, h is 2π-periodic, since for each

Consequently, by what we proved in part (a) it follows that h(x) = e ik•x for some k ∈ Z, and, as a result, f (x) = e iα•x for each x ∈ R, where α = k + β.

Remark 2. The above results and proofs can be generalized almost word for word to characters defined on the n-dimensional flat torus T n and on R n . Now in the proof of Theorem 1(a) we consider periodic functions whose n periods are 2πe j , for each element e j , j = 1, . . . , n, of the (standard) basis, and in the proof of Theorem 1(b) we take β = (β j ) n j=1 ∈ [0, 1) n satisfying e i2πβ j = f (2πe j ) for each j. Remark 3. The above results, although they seem more general than the results described in the references, are in fact equivalent to them, since every measurable character defined on a locally compact group is continuous, as implied by [4,Theorem 22.18,p. 346]. However, the proof of this theorem is quite complicated, and in the cases considered in this note our proofs seem simpler.

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