The most well known and well studied function space is the Hardy space $H^p$. Its first brick was laid by Godfrey Harold Hardy in a paper in the Proceedings of the London Mathematical Society, in which he affirmatively answered a question of Landau [3]. In this work, Hardy generalized Hadamard's three-circle theorem which led to the creation of Hardy spaces. Moreover, for three decades afterwards Hardy, alone or in collaboration with others, discovered many fascination properties of these spaces.

The theory of Hardy spaces has close connections to many branches of mathematics, including Fourier analysis, harmonic analysis, signal and image processing, control theory, singular integrals, probability theory and operator theory. In this introductory chapter, we briefly discuss some representation theorems for the Hardy spaces of the open unit disc. We cover the following topics for function spaces on the open unit disc $\mathbb{D}$:

- Convolution

- Young's inequality

- Approximate identities

- Poisson Kernel

- Fejer Kernel

- The Banach algebra $H^\infty(\mathbb{D})$

- Four equivalent definitions for the Hardy-Hilbert space $H^2(\mathbb{D})$

- $H^p(\mathbb{D})$ as a Banach space

- Construction of $h^1(\mathbb{D})$ functions

- Uniform convergence of continuous functions

- Weak* convergence of measures

- Convergence in $L^p$-norm

- Weak* convergence of $L^\infty$-functions

- Convergence in $L^2$-norm

- Parseval identity

- Poisson representation in $h(\overline{\mathbb{D}})$

- Poisson representation in $h^\infty(\mathbb{D})$

- Poisson representation in $h^p(\mathbb{D})$, $(1<p<\infty)$

- Poisson representation in $h^1(\mathbb{D})$

- Poisson representation in $H^p(\mathbb{D})$, $1 < p \leq \infty$

- Poisson representations in $H^1(\mathbb{D})$

Similar results, albeit somehow more technical, exist for Hardy spaces of the upper half plane. They are collected in the last section. There are excellent books on Hardy spaces: Duren [1], Garnett [2], Koosis [4] and Martınez-Avendano-Rosenthal [5].

Bibliography

[1] Duren, P. L. Theory of Hp spaces. Pure and Applied Mathematics, Vol.38. Academic Press, New York-London, 1970.

[2] Garnett, J. B. Bounded analytic functions, first ed., vol. 236 of Graduate Texts in Mathematics. Springer, New York, 2007.

[3] Hardy, G. H. The mean value of the modulus of an analytic function. Proc. London Math. Soc. 14 (1915), 269–277.

[4] Koosis, P. Introduction to $H^p$ spaces, second ed., vol. 115 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin].

[5] Martınez-Avendano, R. A., and Rosenthal, P. ˜ An introduction to operators on the Hardy-Hilbert space, vol. 237 of Graduate Texts in Mathematics. Springer, New York, 2007.

[6] Mashreghi, J. Representation theorems in Hardy spaces, vol. 74 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2009.