Spherical Harmonics Expansion convergence

In summary, the conversation discusses the expansion of a square-integrable function as a linear combination of Spherical Harmonics in the context of ##L^2## space. This expansion holds in the sense of mean-square convergence, but it does not necessarily hold pointwise. However, if the function is bounded and continuous, it will converge pointwise and in any ##L^p## class. The proof and formal statement of this theorem is not provided, but it is based on recollection of course material. The question of when the expansion converges outside of ##H## is dependent on specific conditions and may not always make sense.
  • #1
Coltrane8
Gold Member
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1
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 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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  • #2
L2 convergence does not preclude pointwise convergence. If the function is bounded and continuous,it will converge pointwise and Lp for any ##p\ge 1##.
 
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  • #3
mathman said:
L2 convergence does not preclude pointwise convergence. If the function is bounded and continuous,it will converge pointwise and Lp for any ##p\ge 1##.

Ok, where can I find this result/theorem? Because it seems to be pretty strong if for any basis in ##L^2##. I know that for spherical harmonic expansion, you would need the function to be ##C^1##, so it seems just not true to me what you affirmed.

Moreover when you talk of a basis of ##L^2##, you are dealing with class of function. In fact the expansions of two different functions, differing in a set of measure zero, is the same. So there exist only one function in the class of function differing by a set of measure zero, that suffice to converge pointwise?

I bet that in the last case we just talk about convergence almost everywhere.
 
Last edited:
  • #4
Bounded and continuous functions on a set of finite measure belong to all Lp classes (trivially obvious). Also two continuous function which are equal almost everywhere are identical.
 
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  • #5
Ok, and this I assume is valid for any ##C^1## or bounded and continuous function expanded on any complete set of orthonormal basis of a Hilbert space. I would like to see the proof and the formal statement of the theorem. May you provide some references were I can look to?

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##?, for example uniformly or pointwise?
 
  • #6
##C^1## is continuous. I don't have a reference. The basis for my statements is my recollection of course material from many years ago.
 
  • #7
Coltrane8 said:
Ok, and this I assume is valid for any ##C^1## or bounded and continuous function expanded on any complete set of orthonormal basis of a Hilbert space. I would like to see the proof and the formal statement of the theorem. May you provide some references were I can look to?

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##?, for example uniformly or pointwise?
You have to be more specific, otherwise the question doesn't always make sense. For example pointwise is meaningless in ##L^2## because the value of a function in ##L^2## makes no sense.
 

Related to Spherical Harmonics Expansion convergence

1. What is the definition of Spherical Harmonics Expansion convergence?

Spherical Harmonics Expansion convergence is a mathematical technique used to approximate a complex function by representing it as a sum of simpler functions known as spherical harmonics. The convergence of this expansion refers to how closely the approximation matches the original function.

2. How is Spherical Harmonics Expansion convergence calculated?

The convergence of a Spherical Harmonics Expansion is typically calculated by comparing the truncated series of the expansion to the original function. This can be done by measuring the error between the two functions or by calculating the rate at which the series approaches the original function.

3. What factors can affect the convergence of a Spherical Harmonics Expansion?

There are several factors that can affect the convergence of a Spherical Harmonics Expansion, including the degree of the expansion, the complexity of the original function, and the number of terms included in the truncated series. Additionally, numerical errors and round-off errors can also impact the convergence.

4. How can one improve the convergence of a Spherical Harmonics Expansion?

There are a few ways to improve the convergence of a Spherical Harmonics Expansion. One approach is to increase the degree of the expansion, which allows for more complex functions to be represented. Another method is to increase the number of terms included in the truncated series, which can provide a more accurate approximation. Additionally, using more advanced numerical techniques and minimizing round-off errors can also improve convergence.

5. What are the applications of Spherical Harmonics Expansion convergence in science?

Spherical Harmonics Expansion convergence has various applications in science, including in fields such as physics, chemistry, and engineering. It is commonly used to approximate solutions to differential equations, model physical phenomena, and analyze data in spherical coordinates. Additionally, it can also be applied in computer graphics and image processing to represent and manipulate spherical objects.

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