Add getindex for DunklXuDisk and tests

src/rect.jl

@@ -1,3 +1,13 @@
+"""
+    KronPolynomial(A, B, C...)
+
+represents the kronecker product of `A`, `B`, …. In particular, if `K = KronPolynomial(A,B)` and `U` is an infinite
+matrix of coefficients we have
+``
+K[SVector(x,y),:]'DiagTrav(U) == A[x,:]'U*B[y,:]
+``
+"""
+
 struct KronPolynomial{d, T, PP} <: MultivariateOrthogonalPolynomial{d, T}
     args::PP
 end

src/rectdisk.jl

@@ -7,8 +7,10 @@
     β::V
 end
 
+DunklXuDisk{T}(β) where T = DunklXuDisk{T, typeof(β)}(β)
 DunklXuDisk(β::T) where T = DunklXuDisk{float(T), T}(β)
 DunklXuDisk() = DunklXuDisk(0)
+DunklXuDisk{T}() where T = DunklXuDisk{T}(0)
 
 ==(D1::DunklXuDisk, D2::DunklXuDisk) = D1.β == D2.β
 
@@ -18,6 +20,17 @@
 show(io::IO, P::DunklXuDisk) = summary(io, P)
 summary(io::IO, P::DunklXuDisk) = print(io, "DunklXuDisk($(P.β))")
 
+function getindex(P::DunklXuDisk{T}, 𝐱::StaticVector{2}, JR::BlockOneTo) where T
+    x,y = 𝐱
+    n = Int(last(JR))
+    ret = zeros(T, n, n)
+    β = P.β
+    ρ = sqrt(1-x^2)
+    for j = 1:n
+        ret[1:n-j+1,j] = Jacobi{T}(j+β-1/2, j+β-1/2)[x,1:n-j+1] * ρ^(j-1) * Jacobi{T}(β, β)[y/ρ,j]
+    end
+    DiagTrav(ret)
+end
 
 """
     DunklXuDiskWeight(β)
@@ -37,6 +50,11 @@
 show(io::IO, P::DunklXuDiskWeight) = summary(io, P)
 summary(io::IO, P::DunklXuDiskWeight) = print(io, "(1-x^2-y^2)^$(P.β) on the unit disk")
 
+function getindex(P::DunklXuDiskWeight, 𝐱::StaticVector{2})
+    r = norm(𝐱)
+    (1-r^2)^P.β
+end
+
 const WeightedDunklXuDisk{T} = WeightedBasis{T,<:DunklXuDiskWeight,<:DunklXuDisk}
 
 WeightedDunklXuDisk(β) = DunklXuDiskWeight(β) .* DunklXuDisk(β)

test/test_rectdisk.jl

@@ -1,6 +1,7 @@
 using MultivariateOrthogonalPolynomials, StaticArrays, BlockArrays, BlockBandedMatrices, ArrayLayouts, Base64,
         QuasiArrays, Test, ClassicalOrthogonalPolynomials, BandedMatrices, FastTransforms, LinearAlgebra
-import MultivariateOrthogonalPolynomials: dunklxu_raising, dunklxu_lowering, AngularMomentum
+import MultivariateOrthogonalPolynomials: dunklxu_raising, dunklxu_lowering, AngularMomentum, coefficients, basis
+using ForwardDiff
 
 @testset "Dunkl-Xu disk" begin
     @testset "basics" begin
@@ -13,6 +14,11 @@ import MultivariateOrthogonalPolynomials: dunklxu_raising, dunklxu_lowering, Ang
         @test xy[SVector(0.1,0.2)] == SVector(0.1,0.2)
         @test x[SVector(0.1,0.2)] == 0.1
         @test y[SVector(0.1,0.2)] == 0.2
+
+        ρ = sqrt(1-0.1^2)
+        @test P[SVector(0.1,0.2),1] ≈ 1
+        @test P[SVector(0.1,0.2),Block(2)] ≈ [0.15,0.2]
+        @test P[SVector(0.1,0.2),Block(3)] ≈ [jacobip(2,1/2,1/2,0.1),jacobip(1,3/2,3/2,0.1)*ρ*legendrep(1,0.2/ρ),ρ^2*legendrep(2,0.2/ρ)]
     end
 
     @testset "operators" begin
@@ -28,27 +34,41 @@ import MultivariateOrthogonalPolynomials: dunklxu_raising, dunklxu_lowering, Ang
         @test WP == WP
 
         x, y = coordinates(P)
-
         L = WP \ WQ
         R = Q \ P
 
+        ∂x = Derivative(P, (1,0))
+        ∂y = Derivative(P, (0,1))
+
+        Dx = Q \ (∂x * P)
+        Dy = Q \ (∂y * P)
+
         X = P \ (x .* P)
         Y = P \ (y .* P)
 
-        @test (L * R)[Block.(1:N), Block.(1:N)] ≈ (I - X^2 - Y^2)[Block.(1:N), Block.(1:N)]
 
-        @test (DunklXuDisk() \ WeightedDunklXuDisk(1.0))[Block.(1:N), Block.(1:N)] ≈ (WeightedDunklXuDisk(0.0) \ WeightedDunklXuDisk(1.0))[Block.(1:N), Block.(1:N)]
+        @testset "lowering/raising" begin
+            @test WP[SVector(0.1,0.2),Block.(1:6)]'L[Block.(1:6),Block.(1:4)] ≈ WQ[SVector(0.1,0.2),Block.(1:4)]'
+            @test Q[SVector(0.1,0.2),Block.(1:4)]'R[Block.(1:4),Block.(1:4)] ≈ P[SVector(0.1,0.2),Block.(1:4)]'
 
-        ∂x = Derivative(P, (1,0))
-        ∂y = Derivative(P, (0,1))
+            @test (DunklXuDisk() \ WeightedDunklXuDisk(1.0))[Block.(1:N), Block.(1:N)] ≈ (WeightedDunklXuDisk(0.0) \ WeightedDunklXuDisk(1.0))[Block.(1:N), Block.(1:N)]
+        end
 
-        Dx = Q \ (∂x * P)
-        Dy = Q \ (∂y * P)
 
-        Mx = Q \ (x .* Q)
-        My = Q \ (y .* Q)
+        @testset "jacobi" begin
+            @test (L * R)[Block.(1:N), Block.(1:N)] ≈ (I - X^2 - Y^2)[Block.(1:N), Block.(1:N)]
+            @test P[SVector(0.1,0.2),Block.(1:5)]'X[Block.(1:5),Block.(1:4)] ≈ 0.1P[SVector(0.1,0.2),Block.(1:4)]'
+            @test P[SVector(0.1,0.2),Block.(1:5)]'Y[Block.(1:5),Block.(1:4)] ≈ 0.2P[SVector(0.1,0.2),Block.(1:4)]'
+        end
+
+        @testset "derivatives" begin
+            @test Q[SVector(0.1,0.2),Block.(1:3)]'Dx[Block.(1:3),Block.(1:4)] ≈ [ForwardDiff.gradient(𝐱 -> DunklXuDisk{eltype(𝐱)}(P.β)[𝐱,k], SVector(0.1,0.2))[1] for k=1:10]'
+            Mx = Q \ (x .* Q)
+            My = Q \ (y .* Q)
+
+            A = (Mx * Dy - My * Dx)[Block.(1:N), Block.(1:N)]
+        end
 
-        A = (Mx * Dy - My * Dx)[Block.(1:N), Block.(1:N)]
 
         B = (Q \ P)[Block.(1:N), Block.(1:N)]
 
@@ -89,4 +109,58 @@ import MultivariateOrthogonalPolynomials: dunklxu_raising, dunklxu_lowering, Ang
         @test stringmime("text/plain", DunklXuDisk()) == "DunklXuDisk(0)"
         @test stringmime("text/plain", DunklXuDiskWeight(0)) == "(1-x^2-y^2)^0 on the unit disk"
     end
+
+    @testset "ladder operators" begin
+        x,y = 𝐱 = SVector(0.1,0.2)
+        ρ = sqrt(1-x^2)
+        ρ′ = -x/ρ
+        n,k = 4,2
+        β = 0
+        P = DunklXuDisk(β)
+        K = Block(n+1)[k+1]
+        @test P[𝐱,K] ≈ Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * Legendre()[y/ρ,k+1]
+        @test Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * diff(Legendre())[y/ρ,k+1] ≈ ρ * diff(P,(0,1))[𝐱 ,K]
+        @test diff(Jacobi(k+1/2,k+1/2))[x,n-k+1] * ρ^(k+1) * Legendre()[y/ρ,k+1] ≈ -k*ρ′*P[𝐱,K] + y*ρ′*diff(P,(0,1))[𝐱,K] + ρ * diff(P,(1,0))[𝐱,K]
+
+        dunklxudisk(n, k, β, γ, x, y) = jacobip(n-k, k+β+γ+1/2, k+β+γ+1/2, x)* (1-x^2)^(k/2) * jacobip(k, β, β, y/sqrt(1-x^2))
+
+        A = coefficients(diff(Jacobi(k+1/2,k+1/2)))
+        B = Jacobi(1,1)\diff(Legendre())
+        # M₀₁
+        @test diff(P,(0,1))[𝐱 ,K] ≈ Jacobi(k-1+3/2,k-1+3/2)[x,n-k+1] * ρ^(k-1) * Jacobi(1,1)[y/ρ,k]B[k,k+1] ≈ DunklXuDisk(1)[𝐱,Block(n)[k]]B[k,k+1] ≈ dunklxudisk(n-1,k-1,1,0,x,y)B[k,k+1]
+        # M₀₂
+        @test (k+1)*Legendre()[x,k+1] + (1+x)*diff(Legendre())[x,k+1] ≈ (k+1)*Jacobi(1,0)[x,k+1] #L₄
+        @test (k+1)*Legendre()[y/ρ,k+1] + (1+y/ρ)*diff(Legendre())[y/ρ,k+1] ≈ (k+1)*Jacobi(1,0)[y/ρ,k+1]
+        @test (k+1)*P[𝐱,K] + (1+y/ρ)*ρ *diff(P,(0,1))[𝐱 ,K]  ≈ (k+1)*Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * Jacobi(1,0)[y/ρ,k+1]
+        # M₀₄
+        @test -(1-x)*(k+1)*Legendre()[x,k+1] - (1-x^2)*diff(Legendre())[x,k+1] ≈ 2*(k+1)*Jacobi(-1,0)[x,k+2] #L₄
+        @test -(1-y/ρ)*(k+1)*P[𝐱,K] - (1-y^2/ρ^2)*ρ *diff(P,(0,1))[𝐱 ,K] ≈ 2*(k+1)*Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * Jacobi(-1,0)[y/ρ,k+2]
+        # M₁₀
+        @test diff(Jacobi(k+1/2,k+1/2))[x,n-k+1] ≈ (n+k+2)/2 * Jacobi(k+3/2,k+3/2)[x,n-k]
+        @test 1/ρ * (-k*ρ′*P[𝐱,K] + y*ρ′*diff(P,(0,1))[𝐱,K] + ρ * diff(P,(1,0))[𝐱,K]) ≈ (n+k+2)/2 * Jacobi(k+3/2,k+3/2)[x,n-k] * ρ^k * Legendre()[y/ρ,k+1] ≈ (n+k+2)/2 * dunklxudisk(n-1,k,0,1,x,y)
+        # M₆₀
+        @test (k+1/2)*Jacobi(k+1/2,k+1/2)[x,n-k+1] + (1+x)*diff(Jacobi(k+1/2,k+1/2))[x,n-k+1] ≈ (n+1/2)*Jacobi(k+3/2,k-1/2)[x,n-k+1] # L₆
+        @test (k+1/2)*Jacobi(k+1/2,k+1/2)[x,n-k+1]* ρ^k * Legendre()[y/ρ,k+1] + (1+x)*diff(Jacobi(k+1/2,k+1/2))[x,n-k+1]* ρ^k * Legendre()[y/ρ,k+1] ≈ (n+1/2)*Jacobi(k+3/2,k-1/2)[x,n-k+1]* ρ^k * Legendre()[y/ρ,k+1]
+        @test (k+1/2)*P[𝐱,K] + (1+x)/ρ*( -k*ρ′*P[𝐱,K] + y*ρ′*diff(P,(0,1))[𝐱,K] + ρ * diff(P,(1,0))[𝐱,K]) ≈ (n+1/2)*Jacobi(k+3/2,k-1/2)[x,n-k+1] * ρ^k * Legendre()[y/ρ,k+1]
+        # M₀₁'
+        @test -(1-x^2)*diff(Legendre())[x,k+1] ≈ 2*(k+1)*Jacobi(-1+eps(),-1+eps())[x,k+2] #L₁'
+        @test -(1-y^2/ρ^2)*diff(Legendre())[y/ρ,k+1] ≈ 2*(k+1)*Jacobi(-1+eps(),-1+eps())[y/ρ,k+2]
+        @test -(1-y^2/ρ^2)* ρ^2 * diff(P,(0,1))[𝐱 ,K] ≈ 2*(k+1)*Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^(k+1) *Jacobi(-1+eps(),-1+eps())[y/ρ,k+2] ≈ 2*(k+1)*dunklxudisk(n+1,k+1,-1+eps(),0,x,y)
+        # M₀₂'
+        @test (1-x)*k*Legendre()[x,k+1] - (1-x^2)*diff(Legendre())[x,k+1] ≈ 2*k*Jacobi(-1,0)[x,k+1] #L₂'
+        @test (1-y/ρ)*k*Legendre()[y/ρ,k+1] - (1-y^2/ρ^2)*diff(Legendre())[y/ρ,k+1] ≈ 2*k*Jacobi(-1,0)[y/ρ,k+1]
+        @test (1-y/ρ)*k*Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * Legendre()[y/ρ,k+1] - (1-y^2/ρ^2)*Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * diff(Legendre())[y/ρ,k+1] ≈ 2*k*Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * Jacobi(-1,0)[y/ρ,k+1] #L₂'
+        @test (1-y/ρ)*k*P[𝐱,K] - (1-y^2/ρ^2)*ρ *diff(P,(0,1))[𝐱 ,K] ≈ 2*k*Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * Jacobi(-1,0)[y/ρ,k+1]
+        # M₀₄'
+        @test -k*Legendre()[x,k+1] + (1+x)*diff(Legendre())[x,k+1] ≈ k*Jacobi(1,0)[x,k] #L₄'
+        @test -k*P[𝐱,K] + (1+y/ρ)*ρ *diff(P,(0,1))[𝐱 ,K] ≈ k*Jacobi(k+1/2,k+1/2)[x,n-k+1] * ρ^k * Jacobi(1,0)[y/ρ,k]
+        # M₁₀'
+        @test ((1+x)*(n-k+1/2) - (1-x)*(n-k+1/2))*Jacobi(k+1/2,k+1/2)[x,n-k+1] - (1-x^2)*diff(Jacobi(k+1/2,k+1/2))[x,n-k+1] ≈ 2*(n-k+1)*Jacobi(k-1/2,k-1/2)[x,n-k+2] # L₁'
+        @test ((1+x)*(n-k+1/2) - (1-x)*(n-k+1/2))*P[𝐱,K] - (1-x^2)/ρ * (-k*ρ′*P[𝐱,K] + y*ρ′*diff(P,(0,1))[𝐱,K] + ρ * diff(P,(1,0))[𝐱,K]) ≈ 2*(n-k+1)*Jacobi(k-1/2,k-1/2)[x,n-k+2]* ρ^k * Legendre()[y/ρ,k+1]
+        # M₆₀'
+        @test (k+1/2)*Jacobi(k+1/2,k+1/2)[x,n-k+1]-(1-x)*diff(Jacobi(k+1/2,k+1/2))[x,n-k+1] ≈ (n+1/2)*Jacobi(k-1/2,k+3/2)[x,n-k+1] # L₆'
+        @test (k+1/2)*P[𝐱,K]-(1-x)/ρ * (-k*ρ′*P[𝐱,K] + y*ρ′*diff(P,(0,1))[𝐱,K] + ρ * diff(P,(1,0))[𝐱,K]) ≈ (n+1/2)*Jacobi(k-1/2,k+3/2)[x,n-k+1]* ρ^k * Legendre()[y/ρ,k+1]
+
+        # d/dx = M₁₀M₀₂ + M₀₄'M₆₀'
+    end
 end