- for some reason I wasn't following you; I forget why. I don['t want to retoot your toot since it provides minimal context: people will only be interested in it if they are interested in the words
Confusion about hemistpherical harmoni...
It is known (see this answer on physics.se and this one...
However I can probably help with your questions! For item 2, note that both differentiable functions vanishing on the boundary of the hemisphere, and those with vanishing derivative normal to the hemisphere, are dense in 𝐿² of the hemisphere! Thus it's not surprising that both the hemispherical harmonics and complementary hemispherical harmonics are orthonormal bases of 𝐿² of the hemisphere.
The topology of 𝐿² really matters here. Note that a condition like "vanishes on the hemisphere" or "has vanishing derivative normal to the hemisphere" do not define closed subspaces of 𝐿². They define dense subspaces of 𝐿².
You can understand these issues much more easily by thinking about the analogous situation on the interval [0,1].
John Carlos Baez
npub1nf…3nqe4
2025-03-05 16:57:56 UTC
in reply to nevent1q…ujny
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"content": "- for some reason I wasn't following you; I forget why. I don['t want to retoot your toot since it provides minimal context: people will only be interested in it if they are interested in the words\n\nConfusion about hemistpherical harmoni...\nIt is known (see this answer on physics.se and this one...\n\nHowever I can probably help with your questions! For item 2, note that both differentiable functions vanishing on the boundary of the hemisphere, and those with vanishing derivative normal to the hemisphere, are dense in 𝐿² of the hemisphere! Thus it's not surprising that both the hemispherical harmonics and complementary hemispherical harmonics are orthonormal bases of 𝐿² of the hemisphere.\n\nThe topology of 𝐿² really matters here. Note that a condition like \"vanishes on the hemisphere\" or \"has vanishing derivative normal to the hemisphere\" do not define closed subspaces of 𝐿². They define dense subspaces of 𝐿².\n\nYou can understand these issues much more easily by thinking about the analogous situation on the interval [0,1].",
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