XIV

INTRODUCTION

values

u(x, 0) — I/J(X) at t = 0.

The function ip(x) is assumed to be everywhere continuous and

bounded. It being assumed that both u and x/j as functions of x

are periodic of period 1.

From physical considerations based on the superposition principle and the idea of

separation of variables one is lead to the two solutions

u(x,t)= / VK0{l + 2 y -

4 M

c o s 2 ™ ( x - £ ) H

Jo

* • „=i

J

and

1 oo

*o{1^ £ .-~-"*'«H

V

V—

—

OO

Using an elementary lemma from the calculus of variations, the uniqueness of the

solution u, itself a consequence of the energy estimate ^ JQ

u2dx

0, leads imme-

diately to an equivalent form of the theta formula. The functional equation of the

Riemann zeta function in its usual form is then obtained by applying a suitable

Mellin transform to both sides of the theta identity.

New ideas in functional analysis and the rise of the theory of distributions made

it possible to develop the above elementary argument into a powerful technique

capable of new applications to two closely related branches of mathematics: to zeta

and L-functions and to infinite dimensional group representations.

Weil was the first to formulate in the local to global language of Tate's Thesis

the relation between functional equations and the uniqueness of certain zeta distri-

butions. At the heart of his new interpretation is an old idea of Weil concerning

the determination of relative invariant measures, which had already appeared in his

classic work on integration over topological groups. The subsequent full develop-

ment of these ideas by Weil himself, and the light they shed on Siegel's insights into

the Poisson Summation Formula, established beyond doubt the fruitfulness of this

new point of view. Chapter I is an elementary introduction to this circle of ideas.

It includes a detailed presentation of Tate's Thesis together with Weil's ideas about

distributions and zeta functions. In this framework, the local calculations at the

infinite prime are simply an elaboration of the well known results of Hadamard and

M. Riesz on homogeneous distributions. Our elementary presentation can serve as

an introduction to the theories of Sato concerning hyperfunctions defined by com-

plex powers of polynomials and zeta functions on prehomogeneous spaces and to

the theory of the Bernstein polynomial. We have also given a brief outline of the

Jacquet-Godement theory of principal L-functions which is the natural generaliza-

tion to GL(n) of Hecke's L-functions viewed as automorphic objects on GL(1). The

principal results of Chapter I are those of Tate and Weil concerning the uniqueness

of local and global zeta distributions (Lemma 9, Theorem 22) and the functional

equation (Theorem 23).

u(x,t)