ApproxFunOrthogonalPolynomials.ChebyshevType

Chebyshev() is the space spanned by the Chebyshev polynomials

    T_0(x),T_1(x),T_2(x),…

where T_k(x) = cos(k*acos(x)). This is the default space as there exists a fast transform and general smooth functions on [-1,1] can be easily resolved.

ApproxFunOrthogonalPolynomials.GaussWeightMethod

GaussWeight(Hermite(L), L) is a space spanned by exp(-Lx²) * H_k(sqrt(L) * x) where H_k(x)'s are Hermite polynomials.

GaussWeight() is equivalent to GaussWeight(Hermite(), 1.0) by default.

ApproxFunOrthogonalPolynomials.LaguerreType

Laguerre(α) is a space spanned by generalized Laguerre polynomials Lₙᵅ(x) 's on (0, Inf), which satisfy the differential equations

    xy'' + (α + 1 - x)y' + ny = 0

Laguerre() is equivalent to Laguerre(0) by default.

ApproxFunOrthogonalPolynomials.RayType
Ray{a}(c,L,o)

represents a scaled ray (with scale factor L) at angle a starting at c, with orientation out to infinity (o = true) or back from infinity (o = false).

ApproxFunOrthogonalPolynomials.UltrasphericalType

Ultraspherical(λ) is the space spanned by the ultraspherical polynomials

    C_0^{(λ)}(x),C_1^{(λ)}(x),C_2^{(λ)}(x),…

Note that λ=1 this reduces to Chebyshev polynomials of the second kind: C_k^{(1)}(x) = U_k(x). For λ=1/2 this also reduces to Legendre polynomials: C_k^{(1/2)}(x) = P_k(x).