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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 (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 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. Hardy spaces for the unit disk 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. More generally, the Hardy space Hp for 0 < p < ∞ is the class of holomorphic functions f on the open unit disk satisfying sup 0 ⩽ r < 1 ( 1 2 π ∫ 0 2 π | f ( r e i θ ) | p d θ ) 1 p < ∞ . {\displaystyle \sup _{0\leqslant r<1}\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{p}\;\mathrm {d} \theta \right)^{\frac {1}{p}}<\infty .} This class Hp is a vector space. The number on the left side of the above inequality is the Hardy space p-norm for f, denoted by ‖ f ‖ H p . {\displaystyle \|f\|_{H^{p}}.} It is a norm when p ≥ 1, but not when 0 < p < 1. The space H∞ is defined as the vector space of bounded holomorphic functions on the disk, with the norm ‖ f ‖ H ∞ = sup | z | < 1 | f ( z ) | . {\displaystyle \|f\|_{H^{\infty }}=\sup _{|z|<1}\left|f(z)\right|.} For 0 < p ≤ q ≤ ∞, the class Hq is a subset of Hp, and the Hp-norm is increasing with p (it is a consequence of Hölder's inequality that the Lp-norm is increasing for probability measures, i.e. measures with total mass 1). Hardy spaces on the unit circle The Hardy spaces defined in the preceding section can also be viewed as certain closed vector subspaces of the complex Lp spaces on the unit circle. This connection is provided by the following theorem (Katznelson 1976, Thm 3.8): Given f ∈ Hp, with p ≥ 1, the radial limit f ~ ( e i θ ) = lim r → 1 f ( r e i θ ) {\displaystyle {\tilde {f}}\left(e^{i\theta }\right)=\lim _{r\to 1}f\left(re^{i\theta }\right)} exists for almost every θ. The function f ~ {\displaystyle {\tilde {f}}} belongs to the Lp space for the unit circle, and one has that ‖ f ~ ‖ L p = ‖ f ‖ H p . {\displaystyle \|{\tilde {f}}\|_{L^{p}}=\|f\|_{H^{p}}.} Denoting the unit circle by T, and by Hp(T) the vector subspace of Lp(T) consisting of all limit functions f ~ {\displaystyle {\tilde {f}}} , when f varies in Hp, one then has that for p ≥ 1,(Katznelson 1976) g ∈ H p ( T )  if and only if  g ∈ L p ( T )  and  g ^ ( n ) = 0  for a.... Discover the Benjamin P Hardy popular books. Find the top 100 most popular Benjamin P Hardy books.