Hardy space

E451925

Banach space Hilbert space concept in complex analysis function space

A Hardy space is a function space in complex analysis consisting of holomorphic functions on a domain whose mean values on boundary circles (or lines) are uniformly bounded, playing a central role in harmonic and operator theory.

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Observed surface forms (1)

Surface form	Occurrences
Hardy spaces	2

Statements (55)

Predicate	Object
instanceOf	Banach space ⓘ Hilbert space ⓘ concept in complex analysis ⓘ function space ⓘ
characterizedBy	boundary values in L^p ⓘ bounded p-means on circles or lines approaching boundary ⓘ
consistsOf	holomorphic functions ⓘ
definedOn	real line boundary ⓘ unit circle boundary ⓘ unit disk ⓘ upper half-plane ⓘ
domainOfParameter	0 < p ≤ ∞ ⓘ
field	complex analysis ⓘ functional analysis ⓘ harmonic analysis ⓘ operator theory ⓘ
generalizedTo	Hardy spaces on R^n ⓘ Hardy spaces on domains in ℂ^n ⓘ
H^1DualIs	BMO (bounded mean oscillation) ⓘ
H^2Is	Hilbert space ⓘ
H^pIs	Banach space for 1 ≤ p ≤ ∞ ⓘ quasi-Banach space for 0 < p < 1 ⓘ
H^∞PredualIs	H^1 / H^1_0 (up to standard identifications) ⓘ
hasBoundaryValues	non-tangential limits almost everywhere ⓘ
hasDecomposition	inner-outer factorization of H^p functions ⓘ
hasDualSpace	H^q for 1 < p < ∞ with 1/p + 1/q = 1 ⓘ
hasFamily	H^p spaces ⓘ
hasNorm	L^p norm of boundary values ⓘ supremum of L^p means on circles or lines ⓘ
hasOperator	Hankel operator ⓘ Toeplitz operator NERFINISHED ⓘ shift operator ⓘ
hasProperty	analytic continuation inside domain ⓘ closed under pointwise addition ⓘ closed under scalar multiplication ⓘ shift-invariant under multiplication by z ⓘ
namedAfter	G. H. Hardy NERFINISHED ⓘ
parameter	p ⓘ
relatedConcept	Bergman space ⓘ Blaschke product NERFINISHED ⓘ Dirichlet space NERFINISHED ⓘ Fourier series NERFINISHED ⓘ Nevanlinna class NERFINISHED ⓘ Poisson integral ⓘ inner-outer factorization ⓘ outer function ⓘ
specialCase	H^1 ⓘ H^2 ⓘ H^∞ ⓘ
usedIn	Hardy space of the unit disk H^p(D) ⓘ Hardy space of the upper half-plane H^p(ℂ_+) ⓘ control theory ⓘ model theory of contractions ⓘ prediction theory of stationary processes ⓘ signal processing ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Littlewood–Paley theory → appliesTo → Hardy space ⓘ

this entity surface form: Hardy spaces

Hardy → knownFor → Hardy space ⓘ

subject surface form: G. H. Hardy

Godfrey → notableFor → Hardy space ⓘ

subject surface form: G. H. Hardy

Carathéodory–Fejér interpolation → relatedTo → Hardy space ⓘ

this entity surface form: Hardy spaces