5 - The Fourier Analysis Point of View

Summary

We work with integrable functions on the polytorus, both in finite and infinitely many variables. For such a function and a multi-index the corresponding Fourier coefficient is defined. For each 1 ≤ p ≤ ∞ the Hardy space H_p consists of those functions in L_p having non-zero Fourier coefficients only for multi-indices in the positive cone. The Hardy space H_\infty on the infinite dimensional polytorus and the space of bounded holomorphic functions on Bc0 are isometrically isomorphic. To prove this the Poisson kernel in several variables is defined, and the Poisson operator (defined through convolution with this kernel) is considered. With these it is shown that the trigonometric polynomials are dense in L_p for 1 ≤ p < ∞ and weak*-dense in L_\infty, and that so also are the analytic trigonometric polynomials in H_p and H_∞. The isometry between the two spaces is first established for the finite dimensional polytorus/polydisc and then, using a version of Hilbert’s criterion (see Chapter 2), raised to the infinite-dimensional case. The density of the polynomials can be proved using the Féjer kernel instead of the Poisson one.