ApproxFun+IntervalArithmetic: Support evaluation with infinite vectors

[dlfivefifty]

It's possible to represent functions with infinite coefficients using InfiniteArrays.jl, however, evaluation is broken:

julia> f = Fun(Chebyshev(), -2.0 .^(-(1:∞)))
Fun(Chebyshev(),[-0.5, -0.25, -0.125, -0.0625, -0.03125, -0.015625, -0.0078125, -0.00390625, -0.00195313, -0.000976563  …  ])

julia> f(0.1)
ERROR: ArgumentError: Cannot create range starting at infinity
Stacktrace:
 [1] (::Colon)(::InfiniteArrays.Infinity, ::Int64, ::Int64) at /Users/solver/Projects/InfiniteArrays.jl/src/infrange.jl:10
 [2] chebyshev_clenshaw(::BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(-),Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}}, ::Float64) at /Users/solver/Projects/ApproxFunBase.jl/src/LinearAlgebra/clenshaw.jl:243
 [3] clenshaw(::Chebyshev{ChebyshevInterval{Float64},Float64}, ::BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(-),Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}}, ::Float64) at /Users/solver/Projects/ApproxFunOrthogonalPolynomials.jl/src/Spaces/Chebyshev/Chebyshev.jl:73
 [4] evaluate at /Users/solver/Projects/ApproxFunOrthogonalPolynomials.jl/src/Spaces/PolynomialSpace.jl:23 [inlined]
 [5] evaluate at /Users/solver/Projects/ApproxFunBase.jl/src/Fun.jl:216 [inlined]
 [6] (::Fun{Chebyshev{ChebyshevInterval{Float64},Float64},Float64,BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(-),Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}}})(::Float64) at /Users/solver/Projects/ApproxFunBase.jl/src/Fun.jl:220
 [7] top-level scope at none:0

It's actually a bit simpler to think about in the interval arithmetic setting where we only need a bound on evaluation (otherwise we would need to talk about tolerances). That is, in the format that #2 would return:

julia> f = Fun(Chebyshev(), BroadcastArray(IntervalArithmetic.Interval, -2.0 .^(-(1:∞)), 2.0 .^(-(1:∞))))
Fun(Chebyshev(),IntervalArithmetic.Interval{Float64}[[-0.5, 0.5], [-0.25, 0.25], [-0.125, 0.125], [-0.0625, 0.0625], [-0.03125, 0.03125], [-0.015625, 0.015625], [-0.0078125, 0.0078125], [-0.00390625, 0.00390625], [-0.00195313, 0.00195313], [-0.000976563, 0.000976563]  …  ])

julia> f(0.1)
ERROR: ArgumentError: Cannot create range starting at infinity
Stacktrace:
 [1] (::Colon)(::InfiniteArrays.Infinity, ::Int64, ::Int64) at /Users/solver/Projects/InfiniteArrays.jl/src/infrange.jl:10
 [2] chebyshev_clenshaw(::BroadcastArray{IntervalArithmetic.Interval{Float64},1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},Type{IntervalArithmetic.Interval},Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(-),Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}},BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}}, ::Float64) at /Users/solver/Projects/ApproxFunBase.jl/src/LinearAlgebra/clenshaw.jl:243
 [3] clenshaw(::Chebyshev{ChebyshevInterval{Float64},Float64}, ::BroadcastArray{IntervalArithmetic.Interval{Float64},1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},Type{IntervalArithmetic.Interval},Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(-),Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}},BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}}, ::Float64) at /Users/solver/Projects/ApproxFunOrthogonalPolynomials.jl/src/Spaces/Chebyshev/Chebyshev.jl:73
 [4] evaluate at /Users/solver/Projects/ApproxFunOrthogonalPolynomials.jl/src/Spaces/PolynomialSpace.jl:23 [inlined]
 [5] evaluate at /Users/solver/Projects/ApproxFunBase.jl/src/Fun.jl:216 [inlined]
 [6] (::Fun{Chebyshev{ChebyshevInterval{Float64},Float64},IntervalArithmetic.Interval{Float64},BroadcastArray{IntervalArithmetic.Interval{Float64},1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},Type{IntervalArithmetic.Interval},Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(-),Tuple{BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}},BroadcastArray{Float64,1,Base.Broadcast.Broadcasted{LazyArrays.LazyArrayStyle{1},Tuple{InfiniteArrays.OneToInf{Int64}},typeof(^),Tuple{Float64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}}}})(::Float64) at /Users/solver/Projects/ApproxFunBase.jl/src/Fun.jl:220
 [7] top-level scope at none:0

For Chebyshev it should be easy to work out bounds and override chebyshev_clenshaw(::BroadcastArray{...}, x) to return these bounds. We could also then override chebyshev_clenshaw(a::Vcat, x) to do Clenshaw recursively, using the result of chebyshev_clenshaw(a.arrays[end], x) to determine the initial condition for the rest of Clenshaw. This would give rigorous evaluation of functions.