Expand functions in P'(x .^2 .* P)

src/bases/bases.jl

@@ -145,15 +145,46 @@ function _broadcast_mul_ldiv(::Tuple{ScalarLayout,ApplyLayout{typeof(*)}}, A, B)
     a * (A \ b)
 end
 
-_broadcast_mul_ldiv(::Tuple{ScalarLayout,AbstractBasisLayout}, A, B) =
-    _broadcast_mul_ldiv((ScalarLayout(),UnknownLayout()), A, B)
+_broadcast_mul_ldiv(::Tuple{ScalarLayout,AbstractBasisLayout}, A, B) = _broadcast_mul_ldiv((ScalarLayout(),UnknownLayout()), A, B)
 _broadcast_mul_ldiv(_, A, B) = copy(Ldiv{typeof(MemoryLayout(A)),UnknownLayout}(A,B))
 
 copy(L::Ldiv{<:AbstractBasisLayout,BroadcastLayout{typeof(*)}}) = _broadcast_mul_ldiv(map(MemoryLayout,arguments(L.B)), L.A, L.B)
 copy(L::Ldiv{<:MappedBasisLayouts,BroadcastLayout{typeof(*)}}) = _broadcast_mul_ldiv(map(MemoryLayout,arguments(L.B)), L.A, L.B)
 
 
 
+# multiplication operators, reexpand in basis A
+@inline function _broadcast_mul_adj(::Tuple{Any,AbstractBasisLayout}, Ac, B)
+    a,b = arguments(B)
+    @assert a isa AbstractQuasiVector # Only works for vec .* mat
+    A = Ac'
+    ab = (A * (A \ a)) .* b # broadcasted should be overloaded
+    MemoryLayout(ab) isa BroadcastLayout && return Ac*transform_ldiv(A, ab)
+    Ac*ab
+end
+
+@inline function _broadcast_mul_adj(::Tuple{Any,ApplyLayout{typeof(*)}}, Ac, B)
+    a,b = arguments(B)
+    @assert a isa AbstractQuasiVector # Only works for vec .* mat
+    args = arguments(*, b)
+    *(Ac*(a .* first(args)), tail(args)...)
+end
+
+
+function _broadcast_mul_adj(::Tuple{ScalarLayout,Any}, Ac, B)
+    a,b = arguments(B)
+    a * (Ac*b)
+end
+
+function _broadcast_mul_adj(::Tuple{ScalarLayout,ApplyLayout{typeof(*)}}, Ac, B)
+    a,b = arguments(B)
+    a * (Ac*b)
+end
+
+_broadcast_mul_adj(::Tuple{ScalarLayout,AbstractBasisLayout}, A, B) = _broadcast_mul_adj((ScalarLayout(),UnknownLayout()), A, B)
+_broadcast_mul_adj(_, A, B) = copy(Mul{typeof(MemoryLayout(A)),UnknownLayout}(A,B))
+
+copy(L::Mul{<:AdjointBasisLayout,BroadcastLayout{typeof(*)}}) = _broadcast_mul_adj(map(MemoryLayout,arguments(L.B)), L.A, L.B)
 
 
 """