Ulam stability
E85413
Ulam stability is a concept in the theory of functional equations that studies when approximate solutions imply the existence of exact solutions nearby, forming the basis of what is now called Hyers–Ulam stability.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Hyers–Ulam stability | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
stability concept in functional equations ⓘ |
| appliesTo |
Cauchy functional equation
ⓘ
Jensen inequality ⓘ
surface form:
Jensen functional equation
additive functional equations ⓘ multiplicative functional equations ⓘ quadratic functional equations ⓘ |
| basisFor | Hyers–Ulam stability theory ⓘ |
| characterizedBy | existence of a bound between approximate and exact solutions ⓘ |
| concerns |
error bounds for approximate solutions
ⓘ
stability of functional equations under perturbations ⓘ |
| coreIdea | approximate satisfaction of a functional equation leads to a nearby exact solution ⓘ |
| defines | conditions under which approximate homomorphisms are close to exact homomorphisms ⓘ |
| field |
functional equations
ⓘ
mathematical analysis ⓘ |
| focusesOn |
approximate solutions of functional equations
ⓘ
existence of exact solutions near approximate ones ⓘ |
| generalizedBy | Hyers–Ulam–Rassias stability ⓘ |
| hasApplication |
stability of derivations
ⓘ
stability of group homomorphisms ⓘ stability of isometries ⓘ stability of linear mappings ⓘ stability of ring homomorphisms ⓘ |
| hasVariant | Hyers–Ulam–Rassias stability ⓘ |
| historicalOrigin | a question posed by Stanisław Ulam in 1940 ⓘ |
| implies | continuous dependence of exact solutions on perturbations ⓘ |
| motivated | study of stability of functional equations ⓘ |
| namedAfter |
Stanislaw Ulam
ⓘ
surface form:
Stanisław Ulam
|
| questionFormulation | whether every approximate solution of a functional equation is near an exact solution ⓘ |
| relatedConcept |
Banach space
ⓘ
approximate homomorphism ⓘ functional inequality ⓘ metric space ⓘ stability of homomorphisms ⓘ |
| relatedTo |
Ulam stability
self-linksurface differs
ⓘ
surface form:
Hyers–Ulam stability
error analysis ⓘ perturbation theory ⓘ stability theory in mathematics ⓘ |
| studiedIn |
nonlinear functional analysis
ⓘ
operator theory ⓘ |
| timePeriod | 20th century mathematics ⓘ |
| typicalSetting |
functional equations between Banach spaces
ⓘ
functional equations between normed spaces ⓘ |
| usedIn |
approximation theory
ⓘ
control theory ⓘ dynamical systems ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hyers–Ulam stability