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I want to learn more about the Hardy space $H^2(\mathcal{S})$ on the complex strip $\mathcal{S} = \{ x+iy \in \mathbb{C} ~|~ x,y \in \mathbb{R},~ |y|<1 \}$. In particular, I am interested in it as a reproducing kernel Hilbert space.


I am following this publication by Bakan and Kaijser.

Usually, one defines a Hardy space on the disc as follows: $$ H^2(\mathbb{D}) = \left\{ f \text{ holomorphic } ~\bigg|~ ||f||_{H^2(\mathbb{D})}^2 = \sup_{0 \leq r < 1} \frac{1}{2\pi} \int_{-\pi}^\pi |f(r e^{i\theta})|^2 d\theta < \infty \right\}. $$ The strip is isomorphic to the disc via the following isomorphism: $$ \phi : \mathcal{S} \to \mathbb{D},~ \phi(z) = \tanh\frac{\pi}{4} z ~,~~~~ \phi^{-1} : \mathbb{D} \to \mathcal{S},~ \phi^{-1}(w) = \frac{4}{\pi} \text{artanh}(w) $$ One can define the a Hardy space on the strip as $H^2(\mathcal{S}) = \{ f \text{ holomorphic } ~\Big|~ ||f||_{H^2(\mathcal{S})}^2 := ||f \circ \phi^{-1}||_{H(\mathbb{D})}^2 < \infty \} $. But this definition is not very elegant because one has to integrate $f(4/\pi~ \text{artanh}(w))$.
However, $H^2(\mathcal{S})$ and its norm are equal to the following space and norm: $$ H'^2(\mathcal{S}) = \left\{ f \text{ holomorphic } ~\bigg|~ ||f||_{H'^2(\mathcal{S})}^2 = \sup_{0\leq y < 1} \int_{\mathbb{R}} \frac{|f(t - iy)|^2 + |f(t + iy)|^2}{4 (\cosh \frac{\pi}{2} t + \cos\frac{\pi}{2} y)} dt < \infty \right\} $$ But these are not Hilbert spaces, they do are only Banach and do not have a scalar product. To obtain a Hilbert space from $H^2(\mathbb{D})$, one has to replace $\sup_{0\leq r < 1} \to \lim_{r \to 1}$ and define the scalar product: $$ \langle f, g \rangle_{H^2(\mathbb{D})} = \lim_{r \to 1} \frac{1}{2\pi} \int_{-\pi}^\pi \overline{f(r e^{i\theta})} g(re^{i\theta}) d\theta. $$ My Question: What is the scalar product on $H'^2(\mathcal{S})$?

I suspect that one has to replacing $\sup_{0\leq y < 1} \to \lim_{y\to 1}$ in the norm and that the scalar product is given by replacing $|f(t \mp iy)|^2$ with $\overline{f(t \mp iy)} g(t \mp iy)$. But I do not know how to prove it because I do not understand the equality of $H^2(\mathcal{S})$ and $H'^2(\mathcal{S})$ (which is proven in the publication).

I am also grateful for any references on the topic.

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