Dear esteemed colleagues,
I would like to ask about the main characteristics of the Hardy space H^2(T) and the L^2(T) space given that T represents the unit circle (not the open unit disk).
Also, what is the relationship between the two spaces?
Any help will be much appreciated.
Thank you in advance for all your valuable answers.
H^2(T) is subspace of L^2(T) . Every function f in L^2(T) has Fourier development of the form f= sum f_k e^{ikt} where sum is over integers Z. f in L^2(T) belongs H^2(T) iff f_k =0 for k
Dear Miodrag Mateljević,
Thank you very much for the informative answer.
I appreciate your help.
See https://en.wikipedia.org/wiki/Hardy_space
and for more details
Duren, P. (1970), Theory of Hp-Spaces, Academic Press
Given f in H^p, with p > 0, the radial limit exists for almost every and it defines the boundary function which belongs to the L^p space for the unit circle.
More precisely,
The Hardy spaces can be viewed as certain closed vector subspaces of the complex L^p spaces on the unit circle. This connection is provided by the following theorem (Katznelson 1976, Thm 3.8): Given f in H^p, with p > 0, the radial limit
$\tilde f\left(e^{i\theta}\right) = \lim_{r\to 1} f\left(re^{i\theta}\right)$
exists for almost every \theta. The function ${\tilde {f}}$ belongs to the L^p space for the unit circle.
A relationship between the spaces H^2(T) and L^2(T), attributed by Professor Mateljević, can be extended/generalized to the analogous relationship between the Orlicz space H_{\phi} and the Hardy-Orlicz space L_{\phi} associated to a real non-negative convex stictly nondecreasing convex function (the so-called Orlicz function) \phi(t) on [0,+ \infty) continuous at 0 such that \phi(0) = 0 and \lim_{t\to \+infty}\phi(t)/ t = \lim_{t\to \+infty}\phi(t) =+ \infty (see, e.g., M. Rosenblum and J. Rovnyak, Topics in Hardy Classes and Univalent Functions, Birkhauser, 1994, Chapter 3). In particular, the spaces H^2(T) and L^2(T) are special cases of Hardy-Orlicz space and Orlicz space, respectively, with the associated Orlicz function \phi(t)=exp(t^2)-1.
A “topological reformulation” of a relationship between the spaces $L^2(T)$ and $H^2(T)$ on the unit circle T of the complex plane, attributed by Professor Mateljević, can be given as follows: “$H^2(T)$ is the closure of the space of all analytic polynomials in the Lebesgue space $L^2(T)$”. (The analytic polynomial P is a function of the form $P(e^{it}=\sum_{k=0}^na_k e^{ikt}$, where n is a nonnegative integer, t is a real number and $a_0,...,a_n$ are complex numbers).
Miodrag Mateljević The fact that any function in H^p of the unit disk admits radial limits on the boundary circle almost everywhere is known as Fatou's Lemma. Right?
Arnab Chattopadhyay You are right "The fact that any function in H^p, p in (0,\infty], of the unit disk admits radial limits on the boundary circle almost everywhere is known as Fatou's Lemma".
Set $f_r(z)=f(rz)$.
Then 𝑓𝑟 converges to some function 𝑓^*∈𝐿𝑝(𝑆^1)pointwise almost everywhere and in 𝐿𝑝 norm. There is corresponding statement for any function in the space of harmonic function h^p, p in (1,\infty].
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