In 1932 Norbert Wiener gave a series of lectures on Fourier analysis at the Univer­ sity of Cambridge. One result of Wiener's visit to Cambridge was his well-known text The Fourier Integral and Certain of its Applications; another was a paper by G. H. Hardy in the 1933 Journalofthe London Mathematical Society. As Hardy says in the introduction to this paper, This note originates from a remark of Prof. N. Wiener, to the effect that "a f and g [= j] cannot both be very small". ... The theo­ pair of transforms rems which follow give the most precise interpretation possible ofWiener's remark. Hardy's own statement of his results, lightly paraphrased, is as follows, in which f is an integrable function on the real line and f is its Fourier transform: x 2 m If f and j are both 0 (Ix1e- /2) for large x and some m, then each is a finite linear combination ofHermite functions. In particular, if f and j are x2 x 2 2 2 both O(e- / ), then f = j = Ae- / , where A is a constant; and if one x 2 2 is0(e- / ), then both are null.

1 Euclidean Spaces.- 1.1 Fourier transform on L1(?n).- 1.2 Hermite functions and L2 theory.- 1.3 Spherical harmonics and symmetry properties.- 1.4 Hardy's theorem on ?n.- 1.5 Beurling's theorem and its consequences.- 1.6 Further results and open problems.- 2 Heisenberg Groups.- 2.1 Heisenberg group and its representations.- 2.2 Fourier transform on Hn.- 2.3 Special Hermite functions.- 2.4 Fourier transform of radial functions.- 2.5 Unitary group and spherical harmonics.- 2.6 Spherical harmonics and the Weyl transform.- 2.7 Weyl correspondence of polynomials.- 2.8 Heat kernel for the sublaplacian.- 2.9 Hardy's theorem for the Heisenberg group.- 2.10 Further results and open problems.- 3 Symmetric Spaces of Rank 1.- 3.1 A Riemannian space associated to Hn.- 3.2 The algebra of radial functions on S.- 3.3 Spherical Fourier transform.- 3.4 Helgason Fourier transform.- 3.5 Hecke-Bochner formula for the Helgason Fourier transform.- 3.6 Jacobi transforms.- 3.7 Estimating the heat kernel.- 3.8 Hardy's theorem for the Helgason Fourier transform.- 3.9 Further results and open problems.