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Hardy space information


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.

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Hardy space

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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...

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Hilbert space

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generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs...

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Singular integral operators on closed curves

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become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure...

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Lp space

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indistinguishable. Bochner space – Type of topological space Orlicz space – Type of function space Hardy space – Concept within complex analysis Riesz–Thorin theorem –...

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Function space

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topology all functions, space of pointwise convergence Hardy space Hölder space Càdlàg functions, also known as the Skorokhod space Lip 0 ( R ) {\displaystyle...

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Toeplitz operator

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is the compression of a multiplication operator on the circle to the Hardy space. Let S 1 {\displaystyle S^{1}} be the complex unit circle, with the standard...

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Thomas Hardy

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Thomas Hardy OM (2 June 1840 – 11 January 1928) was an English novelist and poet. A Victorian realist in the tradition of George Eliot, he was influenced...

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Hilbert transform

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{\displaystyle L^{2}(\mathbb {R} ).} The operator U carries the Hardy space H2(T) onto the Hardy space H 2 ( R ) {\displaystyle H^{2}(\mathbb {R} )} . Bedrosian's...

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Poisson kernel

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Hardy space, H p , {\displaystyle H^{p},} and in particular, ‖ u ‖ H p = ‖ f ‖ L p {\displaystyle \|u\|_{H^{p}}=\|f\|_{L^{p}}} Thus, again, the Hardy...

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Progressive function

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versa. The space of progressive functions is sometimes denoted H + 2 ( R ) {\displaystyle H_{+}^{2}(R)} , which is known as the Hardy space of the upper...

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H square

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a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space. In...

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Bounded mean oscillation

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The space of functions of bounded mean oscillation (BMO), is a function space that, in some precise sense, plays the same role in the theory of Hardy spaces...

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List of vector spaces in mathematics

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spaces in abstract mathematics, by Wikipedia page. Banach space Besov space Bochner space Dual space Euclidean space Fock space Fréchet space Hardy space...

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Operator theory

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followed by projection onto the Hardy space) has inspired the study of similar questions on other spaces, such as the Bergman space. The theory of operator algebras...

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List of mathematic operators

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{G}}} are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions. List of...

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Littlewood subordination theorem

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various function spaces of holomorphic functions on the disk. These spaces include the Hardy spaces, the Bergman spaces and Dirichlet space. Let h be a holomorphic...

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Blaschke product

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products were introduced by Wilhelm Blaschke (1915). They are related to Hardy spaces. A sequence of points ( a n ) {\displaystyle (a_{n})} inside the unit...

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Multiplication operator

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which are compressions of multiplication operators on the circle to the Hardy space. A multiplication operator T f {\displaystyle T_{f}} on L 2 ( X ) {\displaystyle...

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Tube domain

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of several real variables (see multidimensional Laplace transform). Hardy spaces on tubes can be defined in a manner in which a version of the Paley–Wiener...

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Jeff Hardy

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Hardy, Hardy, and Krugman: p. 11. Hardy, Hardy, and Krugman: p. 12. Hardy, Hardy, and Krugman: p. 45. Hardy, Hardy, and Krugman: p. 46. Hardy, Hardy,...

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Hardy Boyz

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The Hardys, also known as the Hardy Boyz, are an American professional wrestling tag team consisting of brothers Matt Hardy and Jeff Hardy. They first...

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Banach space

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Fréchet space – A locally convex topological vector space that is also a complete metric space Hardy space – Concept within complex analysis Hilbert space –...

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List of Hardy Boys books

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This is a list of all Hardy Boys books published, by series. In 1979, the Hardy Boys books began to be published by Wanderer Books Simon & Schuster in...

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Donald Sarason

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2017) was an American mathematician whose research topics included Hardy space theory and VMO. As a professor at the University of California, Berkeley...

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Dirac delta function

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D}{\frac {f(\zeta )\,d\zeta }{\zeta -z}}.} Moreover, let H2(∂D) be the Hardy space consisting of the closure in L2(∂D) of all holomorphic functions in D...

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Disk algebra

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}}),} where H∞(D) denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space). When endowed with the pointwise...

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