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In the contex of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where $Y_\ell^m( \theta , \varphi )$ are the Laplace spherical harmonics.

The context here is important because this equality holds only in the sense of the $L^2$-norm.

This expansion holds in the sense of mean-square convergence — convergence in [[Lp space|L2]] of the sphere — which is to say that

$$\lim_{N\to\infty} \int_0^{2\pi}\int_0^\pi \left|f(\theta,\varphi)-\sum_{\ell=0}^N \sum_{m=- \ell}^\ell f_\ell^m Y_\ell^m(\theta,\varphi)\right|^2\sin\theta\, d\theta \,d\varphi = 0.$$

So in general this limit is NOT pointwise? So I can't say that the value at a point of the function equals the value of the expansion at the same point?

If so, why it's usually stated out of the context of the structure of Hilbert space, that a bounded function or a square integrable function on the unit sphere can be expanded with Spherical Harmonics if it's not pointwise? I mean in some context outside the Hilbert space, where I am not interested in their square integral.

Furthermore, if it's not pointwise, but only in the norm, am I allowed to sum term by term two different functions, with two different expansions, like in the quantum scattering problem?

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  • $\begingroup$ The original context would have been pointwise. $L^2$ theory was not devised in a rigorous way until the early part of the 20th century. The integral version of the Cauchy-Schwarz inequality was published around 1860. $\ell^2$ was defined and studied by Hilbert in the early part of the 20th century, around the time of the Lebesgue integral. Separation of variables leading such expansions was known much earlier from the work of Fourier in the early part of the 19th century. $\endgroup$ – Disintegrating By Parts May 14 '20 at 4:35
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As for the reason this expansion is usually not stated in the Hilbert space context, I suspect it is largely cultural. Spherical harmonics are most commonly encountered by physicists and engineers in order to solve specific problems in three-dimensional space, and introductory physics and engineering textbooks typically do not devote a whole lot of time to develop Hilbert space theory. Mathematics textbooks, on the other hand, usually develop Hilbert space theory in full generality, not just for $L^2(\mathbb{R}^3)$ or $L^2(S^2)$. After all, once you have the full machinery of Hilbert spaces, there isn't much to say about spherical harmonics: you just say that they form an orthonormal basis of $L^2(\mathbb{R}^3)$ or $L^2(S^2)$ and you know everything about them from the Hilbert space point of view.

Yes, in general this $L^2$ limit is not pointwise, just like how Fourier series expansions on $L^2([0,1])$ are in general not pointwise limits. However, for the vast majority of functions that you run across in practice, you should be able to show pointwise or even uniform convergence, because most of the time you are going to take spherical harmonic expansions of fairly nice functions. If the coefficients $f_\ell^m$ decay sufficiently rapidly, then you will have uniform convergence, and often you will have ways of knowing the decay of the coefficients (for one, it's tied to the regularity of the functions you're expanding: smoother functions have more rapidly decay coefficients). That's not to say you'll never run into this issue in a physics or engineering context, but often when you do there are still ways around it (whether the physics or engineering literature handles those ways rigorously is a different story).

Lastly, one does not need to be so careful with term by term summation, so long as you're ok with interpreting the results in the Hilbert space context. This is just a general fact about Hilbert spaces: if $f = \sum_n a_n\phi_n$ and $g = \sum_n b_n\phi_n$ are two expansions in a Hilbert space $H$ in terms of an orthonormal basis $(\phi_n)$, then $\sum_n (a_n+b_n)\phi_n$ is a well-defined object in $H$ and can be shown to agree with $f+g$. However, the moment you try to think about pointwise convergence, you return to the previous issue.

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  • $\begingroup$ Ok, so for any expansion on a complete set of orthonormal basis on a Hilbert space, if I want to use this expansion outside the context of Hilbert space (say on a general physical problem), I have to be careful and prove convergence pointwise/uniform. Is there a theorem or a generalization that states exactly this? Say I have a function on $H$ (say $L^2$) with some restriction (bounded, continuous). I have a complete set of function that form an orthonormal basis of the Hilbert space. When can I say that the expansion converges outside $H$? Say uniformly or pointwise? $\endgroup$ – Coltrane8 May 13 '20 at 10:10
  • $\begingroup$ I mean those expansions are massively used on physical problems that are not stated in the contex of hilbert spaces (like some in quantum mechanics), so I look for pointwise convergence or uniform, so that the expansion is valid. $\endgroup$ – Coltrane8 May 13 '20 at 10:14
  • $\begingroup$ I think in quantum mechanics people are genuinely interested in $L^2$ convergence. Since the wavefunction amplitude squared is interpreted as a continuous probability distribution, you in fact rarely care about its pointwise properties: nothing that happens at a single point affects the distributional properties of a continuous probability distribution. $L^2$ convergence is in fact in some ways the more natural notion to consider, because it plays nicely with probabilistic concepts while being typically easiest to verify. $\endgroup$ – Gyu Eun Lee May 13 '20 at 22:38
  • $\begingroup$ (cont.) You shouldn't dismiss $L^2$ as being some artificial mathematical space. It is in fact unwise to insist on understanding everything pointwise in quantum mechanics. $L^2$ contains some very nasty functions that are impossible to understand pointwise, but nevertheless the theory states that all normalized $L^2$ functions are candidates as wavefunctions. Moreover, uniform convergence is less helpful a concept in quantum mechanics than you might think, again because we really care about the distributional properties of a wavefunction rather than their pointwise properties. $\endgroup$ – Gyu Eun Lee May 13 '20 at 22:41
  • $\begingroup$ (cont.) I think the main reason one would care about pointwise and uniform convergence in quantum mechanics would be if you had some specific computational/numeric scheme in mind. But even in that perspective, one also cares about doing numerics with $L^2$ norms and $L^2$ convergence. To answer the first question in the comments: fast decay on the coefficients is enough for spherical harmonics. There is no general theorem for all orthonormal bases and all Hilbert spaces because not all Hilbert spaces arise as function spaces, so the concept of pointwise convergence does not apply. $\endgroup$ – Gyu Eun Lee May 13 '20 at 22:42

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