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examples/subspaceangles.jl

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# # Subspace angles
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# This example considers the angles between neighbouring Laguerre polynomials with a perturbed measure:
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# ```math
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# \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,
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# \cos\theta_n = \frac{\langle L_n, L_{n+k}\rangle}{\|L_n\|_2 \|L_{n+k}\|_2},\quad{\rm for}\quad 0\le n < N-k,
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# ```
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# where the inner product is defined by $\langle f, g\rangle = \int_0^\infty f(x) g(x) x^\beta e^{-x}{\rm\,d}x$.
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#
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# 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:
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# ```math
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# \cos\theta_n = \dfrac{(V^\top V)_{n, n+k}}{\sqrt{(V^\top V)_{n, n}(V^\top V)_{n+k, n+k}}}.
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# \cos\theta_n = \frac{(V^\top V)_{n, n+k}}{\sqrt{(V^\top V)_{n, n}(V^\top V)_{n+k, n+k}}}.
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# ```
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#
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using FastTransforms, LinearAlgebra

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