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Exploring practical possibilities of approximating functions with frames rather than with a basis. The package is heavily inspired by the Chebfun project and the Julia package ApproxFun.
using BasisFunctions, Plots, DomainSets, FrameFun
gr();After choosing a suitable Basis and Domain, any function can be approximated in the resulting frame:
B = Fourier(61) → -1..1
D = -0.5..0.5
f = x->x
F = Fun(f,B,D)
P = plot(F,layout = 2)
plot!(F,f,subplot=2)
The bases support any AbstractFloat subtype, so high precision approximations are straightforward:
B = Fourier(61) → big(-1)..big(1)
F = Fun(f,B,D)
P = plot(F,layout=2)
plot!(F,f,subplot=2)
In higher dimensions, a basis can be any tensorproduct of (scaled) lower dimensional bases:
using StaticArrays
C = Disk(1.0)\Disk(0.3,SVector(0.2, 0.5))
B = (Fourier(31) → -1.3..1.3)^2
f = (x,y)->exp(x+y)
F = Fun(f,B,C)
P = heatmap(F,layout=2,aspect_ratio=1)
plot!(F,f,subplot=2,aspect_ratio=1)
Even fractal domains are not a problem:
B = (Fourier(31) → -1.0..0.35) ⊗ (Fourier(31) → -0.65..0.65)
f = (x,y)->cos(10*x*y)
F = Fun(f, B, mandelbrot())
P = heatmap(F,layout=2,aspect_ratio=1)
plot!(F,f,aspect_ratio=1,subplot=2)
From the Julia REPL, type ] to enter Pkg mode and run
pkg> add DomainSets, BasisFunctions, FrameFun