orthogonalityProperty
P76852
predicate
Indicates that one entity possesses the property of being orthogonal (i.e., at right angles or having zero inner product) with respect to another entity or a given reference.
Observed surface forms (13)
- orthogonalWithRespectTo ×3
- orthogonalityRelation ×3
- areOrthogonalWithRespectTo ×2
- hasOrthogonalityRelation ×2
- orthogonalityCondition ×2
- areOrthogonalOn ×1
- hasOrthogonality ×1
- orthogonalOn ×1
- orthogonalOnInterval ×1
- orthogonalWithRespectToWeight ×1
- orthogonalityIntegral ×1
- orthogonalityWithRespectTo ×1
- satisfyOrthogonalityRelation ×1
Sample triples (25)
| Subject | Object |
|---|---|
| Legendre symbol | sum_{a mod p} (a/p)=0 ⓘ |
| Jacobi polynomials | weight (1-x)^α (1+x)^β on [-1,1] via predicate surface "areOrthogonalWithRespectTo" ⓘ |
| Jacobi polynomials | ∫_{-1}^1 (1-x)^α (1+x)^β P_m^{(α,β)}(x) P_n^{(α,β)}(x) dx = 0 for m ≠ n via predicate surface "hasOrthogonalityRelation" ⓘ |
| Ramanujan’s sum | sum_{n mod q} c_q(n) = 0 for q > 1 via predicate surface "orthogonalityRelation" ⓘ |
| Ramanujan’s sum | sum_{q ≥ 1} c_q(n) c_q(m) / φ(q) converges to a function of gcd(m,n) via predicate surface "orthogonalityRelation" ⓘ |
| Dirichlet characters | sum over n mod N of χ(n) = 0 for non-principal χ ⓘ |
| Dirichlet characters | sum over characters χ of χ(a)overline{χ(b)} = φ(N) if a ≡ b mod N ⓘ |
| Hermite functions | Gaussian weight via predicate surface "orthogonalWithRespectTo" ⓘ |
| Hermite functions | Lebesgue measure with Gaussian weight via predicate surface "orthogonalWithRespectTo" ⓘ |
|
Hermitian forms (work on quadratic forms)
surface form:
Hermitian form
|
x ⟂ y iff h(x,y)=0 via predicate surface "orthogonalityCondition" ⓘ |
| Bernoulli polynomials | not orthogonal on standard intervals with usual weights ⓘ |
| Liouville function | exhibits cancellation in many average sums over n ⓘ |
| Legendre polynomials | interval [-1,1] via predicate surface "areOrthogonalOn" ⓘ |
| Legendre polynomials | weight function w(x)=1 via predicate surface "areOrthogonalWithRespectTo" ⓘ |
| Legendre polynomials | ∫_{-1}^1 P_m(x)P_n(x) dx = 2/(2n+1) δ_{mn} via predicate surface "satisfyOrthogonalityRelation" ⓘ |
| Sturm–Liouville problem | ∫_a^b w(x) y_m(x) y_n(x) dx = 0 for m ≠ n via predicate surface "hasOrthogonalityRelation" ⓘ |
| Chebyshev polynomials of the first kind | w(x) = (1 - x^2)^(-1/2) via predicate surface "orthogonalWithRespectToWeight" ⓘ |
| Chebyshev polynomials of the first kind | [-1,1] via predicate surface "orthogonalOnInterval" ⓘ |
| Chebyshev polynomials of the first kind | \int_{-1}^1 T_m(x) T_n(x) (1-x^2)^{-1/2} dx = 0 for m ≠ n via predicate surface "orthogonalityRelation" ⓘ |
| Gegenbauer polynomials | weight function (1-x^2)^{\lambda-1/2} via predicate surface "orthogonalWithRespectTo" ⓘ |
| Gegenbauer polynomials | [-1,1] via predicate surface "orthogonalOn" ⓘ |
| Gegenbauer polynomials | \int_{-1}^1 (1-x^2)^{\lambda-1/2} C_m^{(\lambda)}(x) C_n^{(\lambda)}(x) dx = 0 for m \neq n via predicate surface "orthogonalityIntegral" ⓘ |
| Gegenbauer polynomials | \lambda > -1/2 via predicate surface "orthogonalityCondition" ⓘ |
| Jack polynomials | Jack inner product depending on α via predicate surface "orthogonalityWithRespectTo" ⓘ |
| Macdonald polynomials | orthogonal with respect to a certain (q,t)-deformed scalar product via predicate surface "hasOrthogonality" ⓘ |