Nevanlinna–Pick interpolation
E506854
Nevanlinna–Pick interpolation is a classical problem in complex analysis and operator theory that seeks analytic functions, typically bounded by one in the unit disk, which match prescribed values at given points.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
interpolation problem
ⓘ
problem in complex analysis ⓘ problem in operator theory ⓘ |
| applicationArea |
control theory
ⓘ
function theory on the unit disk ⓘ signal processing ⓘ system theory ⓘ |
| classicalVersion | scalar Nevanlinna–Pick interpolation ⓘ |
| codomain | unit disk ⓘ |
| conditionOnNodes | interpolation nodes are distinct ⓘ |
| constraint | |f(z)| ≤ 1 on the unit disk ⓘ |
| dataType |
interpolation nodes in the unit disk
ⓘ
target values in the unit disk ⓘ |
| domain | unit disk ⓘ |
| existenceCriterion | positivity of the Pick matrix ⓘ |
| field |
complex analysis
ⓘ
operator theory ⓘ |
| generalization |
matrix-valued Nevanlinna–Pick interpolation
ⓘ
operator-valued Nevanlinna–Pick interpolation ⓘ |
| goal |
characterize existence of bounded analytic interpolants
ⓘ
find analytic functions matching prescribed values at given points ⓘ |
| hasVariant |
Nevanlinna–Pick interpolation in several variables
NERFINISHED
ⓘ
Nevanlinna–Pick interpolation on the upper half-plane NERFINISHED ⓘ |
| historicalPeriod | early 20th century ⓘ |
| involves |
Hilbert space operators
ⓘ
Nevanlinna–Pick kernels NERFINISHED ⓘ positive definite kernels ⓘ |
| namedAfter |
Georg Pick
NERFINISHED
ⓘ
Rolf Nevanlinna NERFINISHED ⓘ |
| relatedConcept |
Carathéodory interpolation
NERFINISHED
ⓘ
Hardy spaces ⓘ Herglotz functions NERFINISHED ⓘ Nevanlinna class NERFINISHED ⓘ Pick theorem NERFINISHED ⓘ Schur functions ⓘ reproducing kernel Hilbert spaces ⓘ |
| solutionMethod |
linear fractional transformations
ⓘ
operator model theory ⓘ realization theory ⓘ |
| solutionSetProperty | solution set is convex in the Schur class ⓘ |
| typeOfConstraint | interpolation with norm bound ⓘ |
| typicalAssumption |
interpolation nodes lie strictly inside the unit disk
ⓘ
target values lie in the closed unit disk ⓘ |
| typicalFunctionClass |
Schur class
ⓘ
bounded analytic functions on the unit disk ⓘ |
| uses | Pick matrix NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.