new example on subspace angles of Laguerre polynomials with a perturbed measure

Author: MikaelSlevinsky

docs/make.jl

@@ -12,6 +12,7 @@ examples = [
     "padua.jl",
     "sphere.jl",
     "spinweighted.jl",
+    "subspaceangles.jl",
     "triangle.jl",
 ]
 
@@ -41,6 +42,7 @@ makedocs(
                         "generated/padua.md",
                         "generated/sphere.md",
                         "generated/spinweighted.md",
+                        "generated/subspaceangles.md",
                         "generated/triangle.md",
                         ],
                     ]

examples/spinweighted.jl

@@ -27,7 +27,6 @@ k = [2/7, 3/7, 6/7]
 r = (θ,φ) -> [sinpi(θ)*cospi(φ), sinpi(θ)*sinpi(φ), cospi(θ)]
 
 # On the tensor product grid, our function samples are:
-
 F = [exp(im*(k⋅r(θ,φ))) for θ in θ, φ in φ]
 
 # We precompute a spin-$0$ spherical harmonic--Fourier plan:

examples/subspaceangles.jl

@@ -0,0 +1,29 @@
+# # Subspace angles
+# This example considers the angles between neighbouring Laguerre polynomials with a perturbed measure:
+# ```math
+# \cos\theta_n = \dfrac{\langle L_n, L_{n+k}\rangle}{\|L_n|_2 \|L_{n+k}\|_2},\quad{\rm for}\quad 0\le n < N-k,
+# ```
+# where the inner product is defined by $\langle f, g\rangle = \int_0^\infty f(x) g(x) x^\beta e^{-x}{\rm\,d}x$.
+#
+# We do so by connecting Laguerre polynomials to the normalized generalized Laguerre polynomials associated with the perturbed measure. It follows by the inner product of the connection coefficients that:
+# ```math
+# \cos\theta_n = \dfrac{(V^\top V)_{n, n+k}}{\sqrt{(V^\top V)_{n, n}(V^\top V)_{n+k, n+k}}}.
+# ```
+#
+using FastTransforms, LinearAlgebra
+
+# The neighbouring index `k` and the maximum degree `N-1`:
+k, N = 1, 11
+
+# The Laguerre connection parameters:
+α, β = 0.0, 0.5
+
+# We precompute a Laguerre--Laguerre plan:
+P = plan_lag2lag(Float64, N, α, β; norm2=true)
+
+# We apply the plan to the identity, followed by the adjoint plan:
+VtV = P*I
+lmul!(P', VtV)
+
+# From this matrix, the angles are recovered from:
+θ = [acos(VtV[n, n+k]/sqrt(VtV[n, n]*VtV[n+k, n+k])) for n in 1:N-k]