Complex analysis | Hardy spaces | Operator theory

Hardy space

In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz, who named them after G. H. Hardy, because of the paper. In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the Lp spaces of functional analysis. For 1 ≤ p ≤ ∞ these real Hardy spaces Hp are certain subsets of Lp, while for p < 1 the Lp spaces have some undesirable properties, and the Hardy spaces are much better behaved. There are also higher-dimensional generalizations, consisting of certain holomorphic functions on tube domains in the complex case, or certain spaces of distributions on Rn in the real case. Hardy spaces have a number of applications in mathematical analysis itself, as well as in control theory (such as H∞ methods) and in scattering theory. (Wikipedia).

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Wiener process | Lebesgue measure | Poisson kernel | Subharmonic function | Absolute value | Functional analysis | Complex analysis | Upper half-plane | G. H. Hardy | Almost all | Frigyes Riesz | Probability space | Schauder basis | Mathematical analysis | Isomorphism | Banach space | Maximal function | Scattering theory | Martingale (probability theory) | H square | Unit disk | Root mean square | Bounded mean oscillation | Doob's martingale inequality | Haar wavelet | Control theory | Real analysis | Hölder's inequality | Doob's martingale convergence theorems | Distribution (mathematics) | Quasinorm | Hilbert transform | Isometry | Subset | Harmonic function | Holomorphic function | Möbius transformation | Blaschke product | Complex number | Dominated convergence theorem | Lp space | Measure (mathematics) | Tube domain | Space (mathematics)