2In this chapter, following [104] and the exposition in [15], we present the spectral analysis of the normalized Fourier transform on (cf. Exercise 2.4.13). In the last two sections, as an application, we recover some classical results in number theory due to Gauss and Schur, including the celebrated law of quadratic reciprocity.

Preliminary results

We will use the notation and convention as in the beginning of Section 2.2.

This way, the normalized Fourier transform is given by

for all and; see Definition 2.4.1.

Similarly, the corresponding inverse Fourier is given by

for all and. Note also that now Proposition 2.4.6.(iv) becomes

Recall (cf. Definition 2.4.14) that for we denote by the function defined by for all.

Proof. (i) and (ii) are just a reformulation of the Fourier inversion formula (Theorem 2.4.2) and the Plancherel formula (Theorem 2.4.3), respectively; they can also be immediately deduced from the orthogonality relations (Proposition 2.3.5).

Proposition 4.1.2

Proof. (i), (ii), and (iii) follow immediately from Lemma 4.1.1 after observing that for all, and.

Theorem 4.1.3 The characteristic polynomial of is given by

Proof. By virtue of Proposition 4.1.2.