Timeline for Spherical harmonics expansion
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 4, 2022 at 23:37 | history | edited | LSpice | CC BY-SA 4.0 |
Name of paper; link to comment
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| Jun 7, 2021 at 3:28 | comment | added | bathalf15320 | @Coltrane8 You do have the (perhaps weak) consolation that if you work in the generalised context of the space of distributions on the sphere then you have everything you might desire (except for pointwise convergence)--the series always convergences in the distributional sense, you can freely manipulate the terms, you can differentiate term by term, .... The price you have to pay is that the corresponding convergence notion is much weaker than any of the classical ones for functions. Only you can know if it is too high. | |
| May 13, 2020 at 14:37 | comment | added | Iosif Pinelis | @Coltrane8 : (i) I am not aware about general results on the uniform convergence of expansions in any orthonormal basis of $L^2$. (ii) For any two sequences $(a_n)$ and $(b_n)$ in any topological vector space, $\lim_n(a_n+b_n)=\lim_n a_n+\lim_n b_n$ if the latter two limits exist; this is just by the definition en.wikipedia.org/wiki/Topological_vector_space of a topological vector space. Of course, $C^1$ and $L^2$ are topological vector spaces. | |
| May 13, 2020 at 10:01 | comment | added | Coltrane8 | 2) when you say $\lim_n(a_n+b_n)=\lim_n a_n+\lim_n b_n$, the definition of limit depends on the topology I suppose. So in this case all is good because we are using the topology inducted by the Hilbert norm on $L^2$ spaces? | |
| May 13, 2020 at 10:00 | comment | added | Coltrane8 | I'm sorry. I'll rephrase: 1) The result stated above, where the expansion converge uniformly is valid for spherical harmonics. There is a more general theorem that says something like that for any expansion on any complete orthonormal set of $L^2$? (Say if I have another function defined on $R$ bounded and continuous, or $C^1$, and I expand it on the set of Hermite polinomials. Can I say it converges uniformly. I'm looking for this kind of generalization) | |
| May 13, 2020 at 0:14 | comment | added | Iosif Pinelis | @Coltrane8 : I don't understand the point of your penultimate comment. Perhaps, you can rephrase/clarify it. As for your latest comment, members of $C^1$ (in contrast with $L^2$) are true functions (rather than equivalence classes). Moreover, if $f\in C^1$ and $g$ differs from $f$ on a nonempty set of measure $0$, then $g$ is not in $C^1$ (and not even in $C$). | |
| May 12, 2020 at 19:51 | comment | added | Coltrane8 | Moreover I'm now still a bit confused, how can I now talk about convergence uniformly if the orthogonal projection doesn't distinguish by function differing for a set of measure $0$ ? Is that the case that the functions in the class of $f$ are not in $C^1$? | |
| May 12, 2020 at 19:48 | comment | added | Coltrane8 | Ok, and the limit is taken in the norm topology of the Hilbert space in general? And the latter result is an application of the Thm in the link above, and in general not applicable for any other orthonormal set (say Bessel functions or Hermite polinomials). I mean there is no generalization and one must treat expansions on a particular set of functions, case to case. | |
| May 12, 2020 at 19:00 | comment | added | Iosif Pinelis | Previous comment continued: In particular, if $f$ and $g$ are in $C^1$, then the expansion for $f+g$ can be obtained by the term-wise addition of the expansions for $f$ and $g$ and, moreover, the expansion for $f+g$ will converge to $f+g$ uniformly and hence everywhere. | |
| May 12, 2020 at 19:00 | comment | added | Iosif Pinelis | Since $\lim_n(a_n+b_n)=\lim_n a_n+\lim_n b_n$ (provided that the latter two limits exist and, say, are finite) and $(f+g)^m_l=f^m_l+g^m_l$, the expansion for $f+g$ can be obtained by the term-wise addition of the expansions for $f$ and $g$ and, moreover, the expansion for $f+g$ will converge to $f+g$ if (and in the same sense as) the expansions for $f$ and $g$ converge to $f$ and $g$, respectively. | |
| May 12, 2020 at 18:47 | comment | added | Coltrane8 | So how to deal with series sums term by term or derivation, for series expansions of classes of functions? I mean what is the formal development for the properties in analogy with usual functions? | |
| May 12, 2020 at 18:34 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |