Embrace view for subspaces

[dlfivefifty]

At the moment the following makes a lazy *, but I think it should be left as a view:

julia> S = LinearSpline{Float64}(0.0:3);

julia> S[:,2:end-1] # Lazy S * Project
QuasiArrays.ApplyQuasiArray{Float64,2,typeof(*),Tuple{Spline{1,Float64},BandedMatrices.BandedMatrix{Int64,FillArrays.Ones{Int64,2,Tuple{Base.OneTo{Int64},Base.OneTo{Int64}}},Base.OneTo{Int64}}}}(*, (Spline{1,Float64}([0.0, 1.0, 2.0, 3.0]), [0 0; 1 0; 0 1; 0 0]))

julia> view(S,:,2:size(S,2)-1) # proposal: returns this
QuasiArrays.SubQuasiArray{Float64,2,Spline{1,Float64},Tuple{Inclusion{Float64,IntervalSets.Interval{:closed,:closed,Float64}},UnitRange{Int64}},false}(Spline{1,Float64}([0.0, 1.0, 2.0, 3.0]), (Inclusion(0.0..3.0), 2:3), 0, 0)

I believe this is what you were asking for @jagot.

The steps to make it work are

• Support lazy multiplication with views
• Fix getindex for lazy multiplication of views
• Lower D*view(S,:,kr) to view(D*S,:,kr)

[jagot]

I guess this is related to this issue; I'm adding restrictions to FiniteDifferencesQuasi.jl, and encounter the same problem as mentioned in #37:

julia> N = 10
10

julia> ρ = 1.0
1.0

julia> R = RadialDifferences(N, ρ)
Radial finite differences basis {Float64} on 0.0..10.5 with 10 points spaced by ρ = 1.0

julia> r = axes(R, 1)
Inclusion(0.0..10.5)

julia> D = Derivative(r)
Derivative{Float64,IntervalSets.Interval{:closed,:closed,Float64}}(Inclusion(0.0..10.5))

julia> R̃ = R[:,3:8]
Radial finite differences basis {Float64} on 0.0..10.5 with 10 points spaced by ρ = 1.0, restricted to basis functions 3..8 ⊂ 1..10

julia> materialize(applied(*, R̃', D, R))
6×10 BandedMatrices.BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}:
  ⋅   -0.533333   0.0        0.514286    ⋅          ⋅          ⋅         ⋅         ⋅         ⋅
  ⋅     ⋅        -0.514286   0.0        0.507937    ⋅          ⋅         ⋅         ⋅         ⋅
  ⋅     ⋅          ⋅        -0.507937   0.0        0.505051    ⋅         ⋅         ⋅         ⋅
  ⋅     ⋅          ⋅          ⋅        -0.505051   0.0        0.503497   ⋅         ⋅         ⋅
  ⋅     ⋅          ⋅          ⋅          ⋅        -0.503497   0.0       0.502564   ⋅         ⋅
  ⋅     ⋅          ⋅          ⋅          ⋅          ⋅        -0.502564  0.0       0.501961   ⋅

julia> apply(*, R̃', D, R)
6×10 BandedMatrices.BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}:
  ⋅   -0.533333   0.0        0.514286    ⋅          ⋅          ⋅         ⋅         ⋅         ⋅
  ⋅     ⋅        -0.514286   0.0        0.507937    ⋅          ⋅         ⋅         ⋅         ⋅
  ⋅     ⋅          ⋅        -0.507937   0.0        0.505051    ⋅         ⋅         ⋅         ⋅
  ⋅     ⋅          ⋅          ⋅        -0.505051   0.0        0.503497   ⋅         ⋅         ⋅
  ⋅     ⋅          ⋅          ⋅          ⋅        -0.503497   0.0       0.502564   ⋅         ⋅
  ⋅     ⋅          ⋅          ⋅          ⋅          ⋅        -0.502564  0.0       0.501961   ⋅

julia> R̃'D*R
6×10 Array{Float64,2}:
 0.0  -0.533333   0.0        0.514286   0.0        0.0        0.0       0.0       0.0       0.0
 0.0   0.0       -0.514286   0.0        0.507937   0.0        0.0       0.0       0.0       0.0
 0.0   0.0        0.0       -0.507937   0.0        0.505051   0.0       0.0       0.0       0.0
 0.0   0.0        0.0        0.0       -0.505051   0.0        0.503497  0.0       0.0       0.0
 0.0   0.0        0.0        0.0        0.0       -0.503497   0.0       0.502564  0.0       0.0
 0.0   0.0        0.0        0.0        0.0        0.0       -0.502564  0.0       0.501961  0.0

My interpretation so far is that this occurs due to fullmaterialize:
https://github.com/JuliaApproximation/QuasiArrays.jl/blob/21d81858e496435176b811148c9b90e0d142219a/src/matmul.jl#L57-L59

Could an alternative approach to the steps you outlined above be to add a trait that decides whether to fullmaterialize or not? In conjunction with JuliaArrays/LazyArrays.jl#91, I would prefer the views to remain attached to the Basis and leave Derivative untouched, i.e. a string of multiplications a*b*c*... should be resolved to a Mul(a,b,c,...) which then gets materialized. Is that a good approach?

[dlfivefifty]

I think this is done.