Blaschke products
E853121
Blaschke product
analytic function
bounded holomorphic function
inner function
object of complex analysis
rational inner function
Blaschke products are bounded analytic functions on the unit disk formed as (finite or infinite) products of Möbius transformations that map the disk to itself, playing a central role in complex analysis and function theory.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Blaschke product | 0 |
| finite Blaschke product | 0 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Blaschke product
ⓘ
analytic function ⓘ bounded holomorphic function ⓘ inner function ⓘ object of complex analysis ⓘ rational inner function ⓘ |
| appearsIn |
factorization of H^p functions
ⓘ
inner–outer factorization ⓘ |
| BlaschkeCondition | ∑ (1 - |a_n|) < ∞ ⓘ |
| boundedBy | 1 on unit disk ⓘ |
| canBe |
finite product
ⓘ
infinite product ⓘ |
| characterizes | bounded analytic functions with given zero set under Blaschke condition ⓘ |
| codomain | complex plane ⓘ |
| convergesUniformlyOn | compact subsets of unit disk ⓘ |
| definedOn | open unit disk ⓘ |
| degree | number of zeros in unit disk counting multiplicity ⓘ |
| extendsMeromorphicallyTo | Riemann sphere NERFINISHED ⓘ |
| factorParameter | a in unit disk ⓘ |
| field |
complex analysis
ⓘ
function theory ⓘ |
| generalizes | simple disk automorphisms ⓘ |
| hasBoundaryValues | modulus 1 almost everywhere on unit circle ⓘ |
| hasGeneralFactorForm | (z-a)/(1-\overline{a}z) ⓘ |
| is |
inner function in H^2
ⓘ
proper holomorphic self-map of unit disk ⓘ |
| isProductOf |
Möbius transformations
ⓘ
disk automorphisms ⓘ |
| isUnimodularOn | unit circle almost everywhere ⓘ |
| mapsSetInto | unit disk ⓘ |
| namedAfter | Wilhelm Blaschke NERFINISHED ⓘ |
| relatedTo |
Nevanlinna class
NERFINISHED
ⓘ
Smirnov class NERFINISHED ⓘ |
| satisfies |
Blaschke condition on zeros for infinite products
ⓘ
|B(z)| ≤ 1 for |z| < 1 ⓘ |
| specialCase | finite Blaschke product ⓘ |
| subsetOf |
Hardy space H^∞
ⓘ
Hardy spaces H^p for 0 < p ≤ ∞ ⓘ |
| usedFor |
Carleson interpolation
NERFINISHED
ⓘ
Nevanlinna–Pick interpolation NERFINISHED ⓘ constructing invariant subspaces of shift operator ⓘ interpolation problems in the unit disk ⓘ model theory of contractions on Hilbert space ⓘ |
| usedIn |
Beurling’s theorem on invariant subspaces of H^2
NERFINISHED
ⓘ
boundary behavior studies of bounded analytic functions ⓘ spectral theory of shift operators ⓘ |
| zeroMultiplicity | encoded by repeated factors ⓘ |
| zeroSet | sequence in unit disk ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.