Hardy Space - Hardy Spaces On The Unit Circle

Hardy Spaces On The Unit Circle

The Hardy spaces defined in the preceding section can also be viewed as certain closed vector subspaces of the complex Lp spaces on the unit circle. This connection is provided by the following theorem (Katznelson 1976, Thm 3.8): Given f ∈ Hp, with p > 0, the radial limit

\tilde f\left(\mathrm{e}^{\mathrm{i}\theta}\right) = \lim_{r\to 1} f\left(r \mathrm{e}^{\mathrm{i}\theta}\right)

exists for almost every θ. The function belongs to the Lp space for the unit circle, and one has that

Denoting the unit circle by T, and by Hp(T) the vector subspace of Lp(T) consisting of all limit functions, when f varies in Hp, one then has that for p ≥ 1,

(Katznelson 1976), where the ĝ(n) are the Fourier coefficients of a function g integrable on the unit circle,

\forall n \in \mathbb{Z}, \ \ \ \hat{g}(n) = \frac{1}{2\pi}\int_0^{2\pi} g\left(\mathrm{e}^{i\phi}\right) \mathrm{e}^{-in\phi} \, \mathrm{d}\phi.

The space Hp(T) is a closed subspace of Lp(T). Since Lp(T) is a Banach space (for 1 ≤ p ≤ ∞), so is Hp(T).

The above can be turned around. Given a function ∈ Lp(T), with p ≥ 1, one can regain a (harmonic) function f on the unit disk by means of the Poisson kernel Pr:

f\left(r \mathrm{e}^{\mathrm{i}\theta}\right)= \frac{1}{2\pi} \int_0^{2\pi} P_r\left(\theta-\phi\right) \tilde f\left(\mathrm{e}^{\mathrm{i}\phi}\right) \, \mathrm{d}\phi, \ \ \ r < 1,

and f belongs to Hp exactly when is in Hp(T). Supposing that is in Hp(T). i.e. that has Fourier coefficients (an)n ∈ Z with an = 0 for every n < 0, then the element f of the Hardy space Hp associated to is the holomorphic function

In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions. Thus, the space H2 is seen to sit naturally inside L2 space, and is represented by infinite sequences indexed by N; whereas L2 consists of bi-infinite sequences indexed by Z.

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