Inner function

In complex analysis, the Hardy spaces (or Hardy classes) H^p are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz (Riesz 1923), who named them after G. H. Hardy, because of the paper (Hardy 1915). 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 L^p spaces of functional analysis. For 1 ≤ p ≤ ∞ these real Hardy spaces H^p are certain subsets of L^p, while for p < 1 the L^p 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.

For spaces of holomorphic functions on the open unit disk, the Hardy space H² consists of the functions f whose mean square value on the circle of radius r remains bounded as r → 1 from below.

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