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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 , 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 R''n'' 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 ''H''2 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 : 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 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 : 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hardy space」の詳細全文を読む スポンサード リンク
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