`ApproxFunOrthogonalPolynomials.Arc`

— Type`Arc(c,r,(θ₁,θ₂))`

represents the arc centred at `c`

with radius `r`

from angle `θ₁`

to `θ₂`

.

`ApproxFunOrthogonalPolynomials.Chebyshev`

— Type`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.GaussWeight`

— Method`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.Hermite`

— Type`Hermite(L)`

represents `H_k(sqrt(L) * x)`

where `H_k`

are Hermite polynomials. `Hermite()`

is equivalent to `Hermite(1.0)`

.

`ApproxFunOrthogonalPolynomials.Jacobi`

— Type`Jacobi(b,a)`

represents the space spanned by Jacobi polynomials `P_k^{(a,b)}`

, which are orthogonal with respect to the weight `(1+x)^β*(1-x)^α`

`ApproxFunOrthogonalPolynomials.Laguerre`

— Type`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.LaguerreWeight`

— Type`LaguerreWeight(α, L, space)`

weights `space`

by `x^α * exp(-L*x)`

.

`ApproxFunOrthogonalPolynomials.LaguerreWeight`

— Method`LaguerreWeight(α, space)`

weights `space`

by `x^α * exp(-x)`

.

`ApproxFunOrthogonalPolynomials.Line`

— Type`Line{a}(c)`

represents the line at angle `a`

in the complex plane, centred at `c`

.

`ApproxFunOrthogonalPolynomials.Ray`

— Type`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.Ultraspherical`

— Type`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)`

.

`ApproxFunOrthogonalPolynomials.WeightedLaguerre`

— Method`WeightedLaguerre(α)`

is the weighted generalized Laguerre space x^α*exp(-x)*L_k^(α)(x).

`ApproxFunOrthogonalPolynomials.WeightedLaguerre`

— Method`WeightedLaguerre()`

is the weighted Laguerre space exp(-x)*L_k(x).