Nice theorem and write up. Already the one dimensional case is something interesting called “The second symmetric derivative”. And I think it might be used to prove the general case directly: If you add up that result for n orthogonal directions then the left hand side is the Laplacian. The right hand side is a sum of a n limits that at first glance seems to depend heavily on the picked direction, but they can’t as the left hand side is independent of the picked directions. We are free to pick arbitrary many different directions and take the average. In the limit it becomes the average over a sphere but the order of limits is wrong, some standard theorem might apply to resolve that?
Ooh, good idea—you’re usually allowed to exchange integrals/sums and limits when you end up wanting to do that, so something like this should probably work.
Nice theorem and write up. Already the one dimensional case is something interesting called “The second symmetric derivative”. And I think it might be used to prove the general case directly: If you add up that result for n orthogonal directions then the left hand side is the Laplacian. The right hand side is a sum of a n limits that at first glance seems to depend heavily on the picked direction, but they can’t as the left hand side is independent of the picked directions. We are free to pick arbitrary many different directions and take the average. In the limit it becomes the average over a sphere but the order of limits is wrong, some standard theorem might apply to resolve that?
Ooh, good idea—you’re usually allowed to exchange integrals/sums and limits when you end up wanting to do that, so something like this should probably work.