Chapter VIII. Fourier Transform in Euclidean Space

Abstract

This chapter develops some of the theory of the $\mathbb{R}^N$ Fourier transform as an operator that carries certain spaces of complex-valued functions on $\mathbb{R}^N$ to other spaces of such functions.

Sections 1–3 give the indispensable parts of the theory, beginning in Section 1 with the definition, the fact that integrable functions are mapped to bounded continuous functions, and various transformation rules. In Section 2 the main results concern $L^1$, chiefly the vanishing of the Fourier transforms of integrable functions at infinity, the fact that the Fourier transform is one-one, and the all-important Fourier inversion formula. The third section builds on these results to establish a theory for $L^2$. The Fourier transform carries functions in $L^1\cap L^2$ to functions in $L^2$, preserving the $L^2$ norm; this is the Plancherel formula. The Fourier transform therefore extends by continuity to all of $L^2$, and the Riesz–Fischer Theorem says that this extended mapping is onto $L^2$. These results allow one to construct bounded linear operators on $L^2$ commuting with translations by multiplying by $L^{\infty}$ functions on the Fourier transform side and then using Fourier inversion; a converse theorem is proved in the next section.

Section 4 discusses the Fourier transform on the Schwartz space, the subspace of $L^1$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. The Fourier transform carries the Schwartz space in one-one fashion onto itself, and this fact leads to the proof of the converse theorem mentioned above.

Section 5 applies the Schwartz space in $\mathbb{R}^1$ to obtain the Poisson Summation Formula, which relates Fourier series and the Fourier transform. A particular instance of this formula allows one to prove the functional equation of the Riemann zeta function.

Section 6 develops the Poisson integral formula, which transforms functions on $\mathbb{R}^N$ into harmonic functions on a half space in $\mathbb{R}^{N+1}$. A function on $\mathbb{R}^N$ can be recovered as boundary values of its Poisson integral in various ways.

Section 7 specializes the theory of the previous section to $\mathbb{R}^1$, where one can associate a “conjugate” harmonic function to any harmonic function in the upper half plane. There is an associated conjugate Poisson kernel that maps a boundary function to a harmonic function conjugate to the Poisson integral. The boundary values of the harmonic function and its conjugate are related by the Hilbert transform, which implements a “$90^{\circ}$ phase shift” on functions. The Hilbert transform is a bounded linear operator on $L^2$ and is of weak type $(1,1)$.