Support for inner product

[juliohm]

Can ApproxFun.jl compute the inner product between two functions in a Hilbert space?

https://discourse.julialang.org/t/packages-defining-functional-data-types/73539

[dlfivefifty]

Yes? sum(f*conj(g)) should work, or maybe dot(f,g)

[juliohm]

@dlfivefifty the first works, but the second gives me a stack overlflow. Maybe a missing method implementation for dot?

[juliohm]

MWE:

using ApproxFun
using LinearAlgebra # for dot

f = Fun(cos, 0..2π)
g = Fun(sin, 0..2π)

dot(f, g)
ERROR: StackOverflowError:
Stacktrace:
 [1] dot(x::Function, y::Function) (repeats 20126 times)

[juliohm]

Also, maybe ApproxFun.jl could reexport \cdot after the method is properly defined? I do that in my packages that define dot product for custom types so that end users don't need to load LinearAlgebra manually.

[dlfivefifty]

The plan is for all linear algebra-like operations to move to ContinuumArrays.jl which ApproxFun.jl will sit on top of to give a function-like syntax: I don’t like the current “mixed metaphor” design, eg the use of sum to mean integrate

[dlfivefifty]

In fact I think dot is supported there, it’s possible depending on your usage ContinuumArrays and ClassicalOrthogonalPolynomials may be better suited

[juliohm]

Can you please clarify the connection between the different packages? I am new to the ecosystem around ApproxFun.jl and thought it was the main umbrella package loading all dependencies for the final end user. That is probably an incorrect understanding now that you pointed out other packages.

[juliohm]

How would you reproduce the example above with ClassicalOrthogonalPolynomials.jl for example?

[dlfivefifty]

At the moment the packages are not connected. ContinuumArrays + ClassicalOrthogonalPolynomials are an in progress redesign that uses linear algebra language. Eg the following might work (don’t have my laptop handy to check)

x = Inclusion(0..2π)
f = cos.(x)
g = sin.(x)
f’g # or dot(f,g)

if that doesn’t work you can be more explicit and specify the basis:

P = legendre(0..2π)
f = P / P \ cos.(x) # expand in mapped legendre 
…

[juliohm]

I will close the issue because it seems something is on the plans. If there is any documentation explaining how the different packages in the ecosystem talk to each other, that would be helpful. I see lots of very cool packages but it is hard to grasp how they are used together.

[juliohm]

I reopened the issue so that it doesn't get lost in the issue tracker. I understand the underlying packages are being refactored to address the Base.dot implementation.