One dimensional universal approximators and tensor product basis

[ChrisRackauckas]

We should make it very easy to use lower dimensional universal approximators. For example,

f(x,p) = p[1] + p[2] * sin(x) + p[3] * sin(2x) + ...

for sine basis, etc. Then take each AbstractOneDimensionalBasis and do things like:

TensorProduct(Fourier(10),Chebyshev(10),Polynomial(6))

for a 3-dimensional tensor product of 10 sine terms, 10 Chebyschev polynomials, and z, z^2, z^3, z^4, z^5, z^6, i.e. R^3 -> R function.

Then

f=Stack(TensorProduct(Fourier(10),Chebyshev(10),Polynomial(6),4)

would have f(x,p) be an R^3 -> R^4 funciton. This can be setup like FastChain to automatically cut up the vector.

[YingboMa]

Should this be a part of ApproxFun? @dlfivefifty

[dlfivefifty]

I would say it would be better to use the quasi-array framework I'm designing to replace the core of ApproxFun, see

https://github.com/JuliaApproximation/ContinuumArrays.jl
https://github.com/JuliaApproximation/OrthogonalPolynomialsQuasi.jl

The benefit of this is everything then uses array terminology: bases are treating as quasi-arrays, for example

axes(Chebyshev()) == (Inclusion(-1..1),1:∞)
axes(Fourier()) == (Inclusion(0..2π),-∞:∞)

Your code above would then become:

A = kron(Fourier()[:,-5:5],Chebyshev()[:,1:10],Polynomial()[:,1:10])
f = vcat(fill(A,4)...)
f[SVector(0.1,0.2,0.5),SVector(1,2,3)] # returns exp(im*0.1)*T_2(0.2)*0.5^3