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.

Read more about this topic:  Hardy Space

Famous quotes containing the words hardy, spaces, unit and/or circle:

    “A whole village-full of sensuous emotion, scattered abroad all the year long, surged here in a focus for an hour. The forty hearts of those waving couples were beating as they had not done since, twelve months before, they had come together in similar jollity. For the time Paganism was revived in their hearts, the pride of life was all in all, and they adored none other than themselves.”
    —Thomas Hardy (1840–1928)

    “through the spaces of the dark
    Midnight shakes the memory
    As a madman shakes a dead geranium.”
    —T.S. (Thomas Stearns)

    “During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner’s jury. In the case of elected bodies the only way of effecting this is by the Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
    —George Bernard Shaw (1856–1950)

    “A beauty is not suddenly in a circle. It comes with rapture. A great deal of beauty is rapture. A circle is a necessity. Otherwise you would see no one. We each have our circle.”
    —Gertrude Stein (1874–1946)