# How to use ApproxFun to multiply differential operators without running into incompatible space issues?
**URL:** https://discourse.julialang.org/t/how-to-use-approxfun-to-multiply-differential-operators-without-running-into-incompatible-space-issues/76481
**Category:** Numerics
**Tags:** approxfun
**Created:** [February 15, 2022, 11:29am UTC](https://discourse.julialang.org/t/how-to-use-approxfun-to-multiply-differential-operators-without-running-into-incompatible-space-issues/76481 "2022-02-15T11:29:36Z")
**Posts on this page:** 3
**Page:** 1
### Author:  [@jishnub](https://discourse.julialang.org/u/jishnub)
#### Post date: [February 15, 2022, 11:29am UTC](https://discourse.julialang.org/t/how-to-use-approxfun-to-multiply-differential-operators-without-running-into-incompatible-space-issues/76481/1 "2022-02-15T11:29:36Z")
I’m trying to compute `(D + x)^2`, where `D` is the derivative operator.
```julia
julia> x = Fun(identity)
Fun(Chebyshev(),[0.0, 1.0])
julia> D = Derivative(Chebyshev())
ConcreteDerivative : Chebyshev() → Ultraspherical(1)
⋅ 1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 2.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 3.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 4.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 5.0 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 6.0 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 7.0 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 8.0 ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 9.0 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱
julia> D2 = D + x;
julia> D2^2
ERROR: AssertionError: spacescompatible(domainspace(ops[k]), rangespace(ops[k + 1]))
```
Directly evaluating `D2^2` errors, seemingly because the range and domain spaces of `D2` differ. However, the conversion from the range space (`Ultraspherical(1)`, or `ChebyshevU`) to the domain space (`Chebyshev`) doesn’t appear to not be defined, so this doesn’t work either:
```julia
julia> Conversion(rangespace(D2), domainspace(D2))
ERROR: Not implemented
Stacktrace:
[1] error(s::String)
@ Base ./error.jl:33
[2] Conversion(A::Ultraspherical{Int64, ChebyshevInterval{Float64}, Float64}, B::Chebyshev{ChebyshevInterval{Float64}, Float64})
@ ApproxFunOrthogonalPolynomials ~/.julia/packages/ApproxFunOrthogonalPolynomials/QcyFy/src/Spaces/Ultraspherical/UltrasphericalOperators.jl:132
[3] top-level scope
@ REPL[127]:1
```
Is there a way to get around this, possibly using some other basis, for which the conversion would be defined? I tried using `Ultraspherical(1/2)` (Legendre) as the domain space instead of Chebyshev, but that doesn’t seem to work either:
```julia
julia> x = Fun(identity, S)
Fun(Ultraspherical(0.5),[-0.0, 1.0, -0.0, -0.0])
julia> D = Derivative(S)
ConcreteDerivative : Ultraspherical(0.5) → Ultraspherical(1.5)
⋅ 1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 1.0 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱
julia> Conversion(rangespace(D2), domainspace(D2))
ERROR: Not implemented
Stacktrace:
[1] error(s::String)
@ Base ./error.jl:33
[2] Conversion(A::Ultraspherical{Int64, ChebyshevInterval{Float64}, Float64}, B::Chebyshev{ChebyshevInterval{Float64}, Float64})
@ ApproxFunOrthogonalPolynomials ~/.julia/packages/ApproxFunOrthogonalPolynomials/QcyFy/src/Spaces/Ultraspherical/UltrasphericalOperators.jl:132
[3] top-level scope
@ REPL[132]:1
```
Maybe I’m doing something wrong here, but my eventual goal is to obtain the operator
```julia
D2^2 = D^2 + 2xD + x^2 + 1
```
---
### Author:  [@dlfivefifty](https://discourse.julialang.org/u/dlfivefifty)
#### Post date: [February 15, 2022, 11:40am UTC](https://discourse.julialang.org/t/how-to-use-approxfun-to-multiply-differential-operators-without-running-into-incompatible-space-issues/76481/2 "2022-02-15T11:40:51Z")
Create the operator with no spaces and then set the space at the end:
```julia
julia> D = Derivative();
julia> x = Fun();
julia> D^2 + 2x*D + x^2 + 1 : Chebyshev()
PlusOperator : Chebyshev() → Ultraspherical(2)
1.1666666666666667 0.0 4.625 … ⋅ ⋅
0.0 0.84375 0.0 ⋅ ⋅
0.08333333333333333 0.0 0.85 ⋅ ⋅
⋅ 0.03125 0.0 0.020833333333333332 ⋅
⋅ ⋅ 0.025 0.0 ⋱
⋅ ⋅ ⋅ … -1.0677083333333333 ⋱
⋅ ⋅ ⋅ 0.0 ⋱
⋅ ⋅ ⋅ 18.084375 ⋱
⋅ ⋅ ⋅ 0.0 ⋱
⋅ ⋅ ⋅ 0.9520833333333334 ⋱
⋅ ⋅ ⋅ … ⋱ ⋱
```
---
### Author:  [@jishnub](https://discourse.julialang.org/u/jishnub)
#### Post date: [February 15, 2022, 12:54pm UTC](https://discourse.julialang.org/t/how-to-use-approxfun-to-multiply-differential-operators-without-running-into-incompatible-space-issues/76481/3 "2022-02-15T12:54:54Z")
Thanks a lot for the swift reply! That’s very helpful. Just wanted to post that the following works as well, with the derivatives computed automatically:
```julia
julia> D = Derivative()
ConcreteDerivative : ApproxFunBase.UnsetSpace() → ApproxFunBase.UnsetSpace()
julia> D + x
PlusOperator : ApproxFunBase.UnsetSpace() → ApproxFunBase.UnsetSpace()
julia> (D + x)^2
TimesOperator : ApproxFunBase.UnsetSpace() → ApproxFunBase.UnsetSpace()
julia> (D + x)^2 : Chebyshev()
TimesOperator : Chebyshev() → Ultraspherical(2)
1.1666666666666667 0.0 4.625 0.0 … ⋅ ⋅ ⋅ ⋅
0.0 0.84375 0.0 6.28125 0.03125 ⋅ ⋅ ⋅
0.08333333333333333 0.0 0.8499999999999999 0.0 0.0 0.025 ⋅ ⋅
⋅ 0.03125 0.0 0.8854166666666666 -1.09375 0.0 0.020833333333333332 ⋅
⋅ ⋅ 0.025 0.0 0.0 -1.0785714285714285 0.0 ⋱
⋅ ⋅ ⋅ 0.020833333333333332 … 14.109375000000002 0.0 -1.0677083333333335 ⋱
⋅ ⋅ ⋅ ⋅ 0.0 16.095238095238095 0.0 ⋱
⋅ ⋅ ⋅ ⋅ 0.940625 0.0 18.084375 ⋱
⋅ ⋅ ⋅ ⋅ 0.0 0.9469696969696969 0.0 ⋱
⋅ ⋅ ⋅ ⋅ 0.0125 0.0 0.9520833333333334 ⋱
⋅ ⋅ ⋅ ⋅ … ⋅ ⋱ ⋱ ⋱
```