Riesz basis
E747349
A Riesz basis is a sequence in a Hilbert space that is complete and behaves like an orthonormal basis up to a bounded, invertible linear transformation, allowing stable and unique expansions of vectors.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
basis in functional analysis
ⓘ
mathematical concept ⓘ |
| appearsIn | monographs on frames and bases in Hilbert spaces ⓘ |
| characterizedBy |
equivalent to an orthonormal basis
ⓘ
image of an orthonormal basis under a bounded invertible operator ⓘ |
| definedOn | Hilbert space ⓘ |
| field |
Hilbert space theory
ⓘ
functional analysis ⓘ |
| generalizationOf | orthonormal basis up to bounded invertible operator ⓘ |
| hasComponent | Riesz basis constants A and B ⓘ |
| hasCondition |
bounded invertible analysis operator
ⓘ
bounded invertible synthesis operator ⓘ satisfies Riesz basis inequalities ⓘ sequence is complete in the Hilbert space ⓘ sequence is minimal ⓘ |
| hasCriterion | Bari theorem for Riesz bases close to orthonormal bases ⓘ |
| hasDual | unique biorthogonal sequence forming a Riesz basis ⓘ |
| hasInequality | A‖c‖² ≤ ‖∑ c_n x_n‖² ≤ B‖c‖² for some 0 < A ≤ B < ∞ ⓘ |
| hasProperty |
Gram matrix defines a bounded invertible operator on ℓ²
ⓘ
allows stable expansions ⓘ allows unique expansions ⓘ biorthogonal dual sequence exists and is unique ⓘ boundedly complete ⓘ complete sequence ⓘ coordinate functionals are continuous ⓘ equivalent norms via coefficient sequences ⓘ expansion coefficients depend continuously on the vector ⓘ not necessarily normalized ⓘ not necessarily orthogonal ⓘ perturbation stable under small bounded perturbations ⓘ topological basis in Hilbert space ⓘ unconditional if and only if equivalent to an orthonormal basis with unconditional convergence ⓘ |
| hasRelation | Riesz basis = complete Riesz sequence ⓘ |
| implies |
Schauder basis property
ⓘ
closed linear span equals whole Hilbert space ⓘ stability of coefficient functionals ⓘ unique coefficient representation for each vector ⓘ |
| namedAfter | Frigyes Riesz NERFINISHED ⓘ |
| relatedTo |
Riesz sequence
ⓘ
Schauder basis ⓘ biorthogonal system ⓘ frame (functional analysis) ⓘ orthonormal basis ⓘ |
| usedIn |
numerical analysis of operator equations
ⓘ
partial differential equations ⓘ signal processing ⓘ spectral theory ⓘ time-frequency analysis ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.