Hardy Spaces — Mathematics Knowledge System

Overview

Hardy spaces H^p (for 0 < p <= infinity) are spaces of holomorphic functions on the unit disk (or upper half-plane) whose boundary values belong to L^p on the circle. They were introduced by Hardy (1915) to study power series and their boundary behaviour. The space H^2 is the most important: it is a Hilbert space consisting of L^2 functions on the circle whose negative Fourier coefficients all vanish. Hardy spaces appear in signal processing, operator theory, and PDE, and are the natural domain for Toeplitz and Hankel operators.

Intuition

An analytic function on the disk is in H^p if its L^p 'radius norms' are uniformly bounded. The key theorem is that H^p functions have boundary values (in a non-tangential limit sense) almost everywhere on the circle, and these boundary values determine the function completely. H^2 is particularly clean: it consists of L^2 functions on the circle that are 'causal' in the sense that all negative Fourier modes vanish — it is the frequency-domain model for one-sided signals.

Formal Definition

Definition

For 0 < p < infinity, the Hardy space H^p(D) of the unit disk D = {|z| < 1} consists of all holomorphic functions f: D -> C satisfying sup_{0<r<1} (1/(2pi) integral_0^{2pi} |f(re^{i theta})|^p d theta)^{1/p} < infinity. For p = infinity, H^inf = bounded analytic functions. The norm is ||f||_{H^p} = sup_r (1/(2pi) integral |f(re^{itheta})|^p)^{1/p}. Equivalently (for p >= 1), H^p can be identified with the closed subspace of L^p(T) consisting of functions whose Fourier coefficients c_n = 0 for n < 0, where T is the unit circle.

\|f\|_{H^p}^p = \sup_{0<r<1} \frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^p\,d\theta < \infty

H^p norm (p < infinity)

H^p(\mathbb{T}) = \left\{ f \in L^p(\mathbb{T}) : \hat{f}(n) = 0 \text{ for all } n < 0 \right\}

Boundary value characterisation

f(z) = \sum_{n=0}^\infty c_n z^n,\quad \|f\|_{H^2}^2 = \sum_{n=0}^\infty |c_n|^2

H^2 as l^2 of Taylor coefficients

Notation

Notation	Meaning
$H^p(\mathbb{D})$	Hardy space on the unit disk
$H^p(\mathbb{T})$	Hardy space as subspace of L^p on the circle
$P_+$	Riesz projection (from L^2 to H^2)
$H^2_-$	Anti-Hardy space: L^2 functions with Fourier modes only n < 0

Theorems

Theorem 14.1: Theorem 14.1 (Fatou's theorem)

Every f in H^p (0 < p <= infinity) has non-tangential boundary limits f*(e^{i theta}) for almost every e^{i theta} on the unit circle, and the boundary function f* lies in L^p(T). Moreover f -> f* is an isometric embedding for p >= 1.

Theorem 14.2: Theorem 14.2 (F. and M. Riesz)

If f in H^1 and f = 0 on a set of positive measure on T, then f = 0 identically. More generally, the zero set of a non-zero H^p function on the boundary has measure zero.

Theorem 14.3: Theorem 14.3 (Inner-outer factorisation)

Every non-zero f in H^p (p > 0) factors uniquely as f = B * S * F where B is a Blaschke product (encodes the zeros), S is a singular inner function (no zeros, |S*| = 1 a.e. on T), and F is an outer function (|F| = exp(P[log|f*|]) where P is the Poisson integral). This is the inner-outer factorisation.

Worked Examples

1
If f(z) = sum_{n=0}^inf c_n z^n is analytic on D, compute the H^2 norm:
$\|f\|_{H^2}^2 = \sup_{0<r<1}\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^2\,d\theta = \sup_{r<1}\sum_{n=0}^\infty |c_n|^2 r^{2n}$
2
By monotone convergence (r -> 1): ||f||_{H^2}^2 = sum_{n=0}^inf |c_n|^2.
$\|f\|_{H^2} = \left(\sum_{n=0}^\infty |c_n|^2\right)^{1/2} = \|(c_n)\|_{\ell^2}$
3
The map f |-> (c_0, c_1, c_2,...) is an isometric bijection H^2 -> l^2(N_0) = l^2({0,1,2,...}).

✓ Answer

H^2(D) is isometrically isomorphic to l^2(N_0): the isomorphism maps f to its sequence of Taylor coefficients, and the H^2 norm equals the l^2 norm of the coefficients.

Practice Problems

Mediumproof writing

Show that H^inf(D) (bounded analytic functions) is a Banach algebra under pointwise multiplication.

Hardfree response

Describe the inner-outer factorisation of f(z) = z (the identity function on D).

Quiz

Summary

H^p(D) consists of analytic functions on the disk with uniformly bounded L^p circle norms.
Boundary values: every H^p function has non-tangential limits a.e. (Fatou), embedding H^p into L^p(T).
H^2 = {f in L^2(T) : c_n = 0 for n < 0}; isometric to l^2 via Taylor coefficients.
F. and M. Riesz: H^1 functions don't vanish on sets of positive measure.
Inner-outer factorisation: f = B * S * F encodes zeros (B), singular boundary behaviour (S), and boundary modulus (F).

References

BookGarnett, J.B. — Bounded Analytic Functions, Springer, 2007
BookRudin, W. — Real and Complex Analysis (3rd ed.), Chapter 17
Websiteen.wikipedia.org