on the limit of the finite representation of harmonics
$begingroup$
Let $Y_n^j, , -nleq j leq n$ be a spherical harmonic. Let $f$ be an even function on the sphere. Then, $f$ can be introduced as
$$
f=sum_{j=-n}^Ahat f(j)Y_n^j,
$$
where $sum_{j=-n}^A|a_j|>0$ and $A$ is the largest integer in $[-n, n]$ such that $hat f(A)neq 0$.
Now, consider spherical polynomial (finite combination of spherical harmonics). We can just take degree of the function $f$, say $f^2$
$$
f^2=sum_{j=-n}^Ahat f(j)^2(Y_n^j)^2+2sum_{-nleq j<lleq A}hat f(j)hat f(l)Y_n^jY_n^l
$$
Question: spherical polynomial is the finite combination of spherical harmonics. Can one take limit there?
taylor-expansion harmonic-analysis spherical-geometry spherical-harmonics
asked Jan 16 at 14:39
user124297user124297
569
$endgroup$
add a comment |
$begingroup$
Let $Y_n^j, , -nleq j leq n$ be a spherical harmonic. Let $f$ be an even function on the sphere. Then, $f$ can be introduced as
$$
f=sum_{j=-n}^Ahat f(j)Y_n^j,
$$
where $sum_{j=-n}^A|a_j|>0$ and $A$ is the largest integer in $[-n, n]$ such that $hat f(A)neq 0$.
Now, consider spherical polynomial (finite combination of spherical harmonics). We can just take degree of the function $f$, say $f^2$
$$
f^2=sum_{j=-n}^Ahat f(j)^2(Y_n^j)^2+2sum_{-nleq j<lleq A}hat f(j)hat f(l)Y_n^jY_n^l
$$
Question: spherical polynomial is the finite combination of spherical harmonics. Can one take limit there?
taylor-expansion harmonic-analysis spherical-geometry spherical-harmonics
asked Jan 16 at 14:39
user124297user124297
569
$endgroup$
$begingroup$
There is something unclear about your question: if $f$ itself is a finite linear combination of spherical harmonics, then that finite linear combination converges pointwise very well, and, of course, we can square both sides of the equality and rearrange. What limit do you mean to ask about? More general functions $f$? Then, yes, there will be non-trivial convergence issues, and non-trivial questions about squaring such an expansion. Can you clarify?
$endgroup$
– paul garrett
Jan 19 at 20:05
$begingroup$
@Paul Garrett: yes, I would like to consider more general function f, where f is any even function on the sphere. thank you.
$endgroup$
– user124297
Jan 20 at 15:19
$begingroup$
@Paul Garrett: there are two questions here. The main one-yes, I would like to consider any even function f on the sphere. Another question is whether there is a representation of giving f through infinite combination of spherical harmonics. Thank you.
$endgroup$
– user124297
Jan 20 at 15:21
add a comment |
$begingroup$
Let $Y_n^j, , -nleq j leq n$ be a spherical harmonic. Let $f$ be an even function on the sphere. Then, $f$ can be introduced as
$$
f=sum_{j=-n}^Ahat f(j)Y_n^j,
$$
where $sum_{j=-n}^A|a_j|>0$ and $A$ is the largest integer in $[-n, n]$ such that $hat f(A)neq 0$.
Now, consider spherical polynomial (finite combination of spherical harmonics). We can just take degree of the function $f$, say $f^2$
$$
f^2=sum_{j=-n}^Ahat f(j)^2(Y_n^j)^2+2sum_{-nleq j<lleq A}hat f(j)hat f(l)Y_n^jY_n^l
$$
Question: spherical polynomial is the finite combination of spherical harmonics. Can one take limit there?
taylor-expansion harmonic-analysis spherical-geometry spherical-harmonics
asked Jan 16 at 14:39
user124297user124297
569
$endgroup$
Let $Y_n^j, , -nleq j leq n$ be a spherical harmonic. Let $f$ be an even function on the sphere. Then, $f$ can be introduced as
$$
f=sum_{j=-n}^Ahat f(j)Y_n^j,
$$
where $sum_{j=-n}^A|a_j|>0$ and $A$ is the largest integer in $[-n, n]$ such that $hat f(A)neq 0$.
Now, consider spherical polynomial (finite combination of spherical harmonics). We can just take degree of the function $f$, say $f^2$
$$
f^2=sum_{j=-n}^Ahat f(j)^2(Y_n^j)^2+2sum_{-nleq j<lleq A}hat f(j)hat f(l)Y_n^jY_n^l
$$
Question: spherical polynomial is the finite combination of spherical harmonics. Can one take limit there?
taylor-expansion harmonic-analysis spherical-geometry spherical-harmonics
taylor-expansion harmonic-analysis spherical-geometry spherical-harmonics
asked Jan 16 at 14:39
user124297user124297
569
asked Jan 16 at 14:39
user124297user124297
569
edited Jan 19 at 19:50
user124297
asked Jan 16 at 14:39
user124297user124297
569
asked Jan 16 at 14:39
user124297user124297
569
asked Jan 16 at 14:39
user124297user124297
569
569
$begingroup$
There is something unclear about your question: if $f$ itself is a finite linear combination of spherical harmonics, then that finite linear combination converges pointwise very well, and, of course, we can square both sides of the equality and rearrange. What limit do you mean to ask about? More general functions $f$? Then, yes, there will be non-trivial convergence issues, and non-trivial questions about squaring such an expansion. Can you clarify?
$endgroup$
– paul garrett
Jan 19 at 20:05
$begingroup$
@Paul Garrett: yes, I would like to consider more general function f, where f is any even function on the sphere. thank you.
$endgroup$
– user124297
Jan 20 at 15:19
$begingroup$
@Paul Garrett: there are two questions here. The main one-yes, I would like to consider any even function f on the sphere. Another question is whether there is a representation of giving f through infinite combination of spherical harmonics. Thank you.
$endgroup$
– user124297
Jan 20 at 15:21
add a comment |
$begingroup$
There is something unclear about your question: if $f$ itself is a finite linear combination of spherical harmonics, then that finite linear combination converges pointwise very well, and, of course, we can square both sides of the equality and rearrange. What limit do you mean to ask about? More general functions $f$? Then, yes, there will be non-trivial convergence issues, and non-trivial questions about squaring such an expansion. Can you clarify?
$endgroup$
– paul garrett
Jan 19 at 20:05
$begingroup$
@Paul Garrett: yes, I would like to consider more general function f, where f is any even function on the sphere. thank you.
$endgroup$
– user124297
Jan 20 at 15:19
$begingroup$
@Paul Garrett: there are two questions here. The main one-yes, I would like to consider any even function f on the sphere. Another question is whether there is a representation of giving f through infinite combination of spherical harmonics. Thank you.
$endgroup$
– user124297
Jan 20 at 15:21
$begingroup$
There is something unclear about your question: if $f$ itself is a finite linear combination of spherical harmonics, then that finite linear combination converges pointwise very well, and, of course, we can square both sides of the equality and rearrange. What limit do you mean to ask about? More general functions $f$? Then, yes, there will be non-trivial convergence issues, and non-trivial questions about squaring such an expansion. Can you clarify?
$endgroup$
– paul garrett
Jan 19 at 20:05
$begingroup$
There is something unclear about your question: if $f$ itself is a finite linear combination of spherical harmonics, then that finite linear combination converges pointwise very well, and, of course, we can square both sides of the equality and rearrange. What limit do you mean to ask about? More general functions $f$? Then, yes, there will be non-trivial convergence issues, and non-trivial questions about squaring such an expansion. Can you clarify?
$endgroup$
– paul garrett
Jan 19 at 20:05
$begingroup$
@Paul Garrett: yes, I would like to consider more general function f, where f is any even function on the sphere. thank you.
$endgroup$
– user124297
Jan 20 at 15:19
$begingroup$
@Paul Garrett: yes, I would like to consider more general function f, where f is any even function on the sphere. thank you.
$endgroup$
– user124297
Jan 20 at 15:19
$begingroup$
@Paul Garrett: there are two questions here. The main one-yes, I would like to consider any even function f on the sphere. Another question is whether there is a representation of giving f through infinite combination of spherical harmonics. Thank you.
$endgroup$
– user124297
Jan 20 at 15:21
$begingroup$
@Paul Garrett: there are two questions here. The main one-yes, I would like to consider any even function f on the sphere. Another question is whether there is a representation of giving f through infinite combination of spherical harmonics. Thank you.
$endgroup$
– user124297
Jan 20 at 15:21
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes, as you anticipate, although the spirit of "every function can be representated as an infinite sum of spherical harmonics", there are problems with convergence, as well as problems with the meaning of multiplication of two infinite sums of functions. These issues already arise (famously) on the one-sphere and the theory of Fourier series.
As a clear simple case: if the function $f$ is smooth, then the infinite sum of spherical harmonics converges pointwise uniformly to it, can be differentiated termwise, and all derivatives' infinite series expansions converge uniformly pointwise as well. And then it is immediate that the series can be squared and express $f^2$.
Similarly, if $f$ is merely sufficiently differentiable, its infinite expression in spherical harmonics converges (absolutely) uniformly pointwise, and can be squared as you indicated.
Details about how much differentiability, of various sorts, suffices are more complicated.
EDIT: For references for rigorous treatment: Stein and Weiss' old book "Introduction Fourier Analysis on Euclidean Spaces" has a chapter that covers many set-up issues about rigorous treatment of spherical harmonics, including things like estimate of sup norms in terms of $L^2$ norms. Also, Wikipedia's page on "spherical harmonics" does include several remarks about convergence arguments. My own course notes from introductory modular forms courses (at http://www.math.umn.edu/~garrett/m/mfms/) include some essays that treat the $L^2$ Sobolev theory, in the context of Weyl's criterion for equidistribution on spheres: http://www.math.umn.edu/~garrett/m/mfms/notes_2014-15/09_spheres.pdf
One basic assertion about convergence via Sobolev theory is the following. For a test function $f$ on the sphere, and non-negative integer $k$, $|f|^2_k=langle (1-Delta)^k f,frangle$ is the $k$th Sobolev norm (squared). Let $H^k$ be the completion of test functions with respect to that norm. Then a basic (not-so-trivial) theorem is that for $k>n/2$ the space $H^k$ lies inside the space of continuous functions, and the convergence of the finite partial sums of the spherical harmonic expansion converge uniformly pointwise to the function.
answered Jan 20 at 15:56
paul garrettpaul garrett
32.1k362120
$endgroup$
$begingroup$
Thank you, Paul. May I ask you to elaborate. At least, some literature and where to start from. I am not familiar with the subject. Thank you.
$endgroup$
– user124297
Jan 21 at 12:20
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3075812%2fon-the-limit-of-the-finite-representation-of-harmonics%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes, as you anticipate, although the spirit of "every function can be representated as an infinite sum of spherical harmonics", there are problems with convergence, as well as problems with the meaning of multiplication of two infinite sums of functions. These issues already arise (famously) on the one-sphere and the theory of Fourier series.
As a clear simple case: if the function $f$ is smooth, then the infinite sum of spherical harmonics converges pointwise uniformly to it, can be differentiated termwise, and all derivatives' infinite series expansions converge uniformly pointwise as well. And then it is immediate that the series can be squared and express $f^2$.
Similarly, if $f$ is merely sufficiently differentiable, its infinite expression in spherical harmonics converges (absolutely) uniformly pointwise, and can be squared as you indicated.
Details about how much differentiability, of various sorts, suffices are more complicated.
EDIT: For references for rigorous treatment: Stein and Weiss' old book "Introduction Fourier Analysis on Euclidean Spaces" has a chapter that covers many set-up issues about rigorous treatment of spherical harmonics, including things like estimate of sup norms in terms of $L^2$ norms. Also, Wikipedia's page on "spherical harmonics" does include several remarks about convergence arguments. My own course notes from introductory modular forms courses (at http://www.math.umn.edu/~garrett/m/mfms/) include some essays that treat the $L^2$ Sobolev theory, in the context of Weyl's criterion for equidistribution on spheres: http://www.math.umn.edu/~garrett/m/mfms/notes_2014-15/09_spheres.pdf
One basic assertion about convergence via Sobolev theory is the following. For a test function $f$ on the sphere, and non-negative integer $k$, $|f|^2_k=langle (1-Delta)^k f,frangle$ is the $k$th Sobolev norm (squared). Let $H^k$ be the completion of test functions with respect to that norm. Then a basic (not-so-trivial) theorem is that for $k>n/2$ the space $H^k$ lies inside the space of continuous functions, and the convergence of the finite partial sums of the spherical harmonic expansion converge uniformly pointwise to the function.
answered Jan 20 at 15:56
paul garrettpaul garrett
32.1k362120
$endgroup$
$begingroup$
Thank you, Paul. May I ask you to elaborate. At least, some literature and where to start from. I am not familiar with the subject. Thank you.
$endgroup$
– user124297
Jan 21 at 12:20
add a comment |
$begingroup$
Yes, as you anticipate, although the spirit of "every function can be representated as an infinite sum of spherical harmonics", there are problems with convergence, as well as problems with the meaning of multiplication of two infinite sums of functions. These issues already arise (famously) on the one-sphere and the theory of Fourier series.
As a clear simple case: if the function $f$ is smooth, then the infinite sum of spherical harmonics converges pointwise uniformly to it, can be differentiated termwise, and all derivatives' infinite series expansions converge uniformly pointwise as well. And then it is immediate that the series can be squared and express $f^2$.
Similarly, if $f$ is merely sufficiently differentiable, its infinite expression in spherical harmonics converges (absolutely) uniformly pointwise, and can be squared as you indicated.
Details about how much differentiability, of various sorts, suffices are more complicated.
EDIT: For references for rigorous treatment: Stein and Weiss' old book "Introduction Fourier Analysis on Euclidean Spaces" has a chapter that covers many set-up issues about rigorous treatment of spherical harmonics, including things like estimate of sup norms in terms of $L^2$ norms. Also, Wikipedia's page on "spherical harmonics" does include several remarks about convergence arguments. My own course notes from introductory modular forms courses (at http://www.math.umn.edu/~garrett/m/mfms/) include some essays that treat the $L^2$ Sobolev theory, in the context of Weyl's criterion for equidistribution on spheres: http://www.math.umn.edu/~garrett/m/mfms/notes_2014-15/09_spheres.pdf
One basic assertion about convergence via Sobolev theory is the following. For a test function $f$ on the sphere, and non-negative integer $k$, $|f|^2_k=langle (1-Delta)^k f,frangle$ is the $k$th Sobolev norm (squared). Let $H^k$ be the completion of test functions with respect to that norm. Then a basic (not-so-trivial) theorem is that for $k>n/2$ the space $H^k$ lies inside the space of continuous functions, and the convergence of the finite partial sums of the spherical harmonic expansion converge uniformly pointwise to the function.
answered Jan 20 at 15:56
paul garrettpaul garrett
32.1k362120
$endgroup$
$begingroup$
Thank you, Paul. May I ask you to elaborate. At least, some literature and where to start from. I am not familiar with the subject. Thank you.
$endgroup$
– user124297
Jan 21 at 12:20
add a comment |
$begingroup$
Yes, as you anticipate, although the spirit of "every function can be representated as an infinite sum of spherical harmonics", there are problems with convergence, as well as problems with the meaning of multiplication of two infinite sums of functions. These issues already arise (famously) on the one-sphere and the theory of Fourier series.
As a clear simple case: if the function $f$ is smooth, then the infinite sum of spherical harmonics converges pointwise uniformly to it, can be differentiated termwise, and all derivatives' infinite series expansions converge uniformly pointwise as well. And then it is immediate that the series can be squared and express $f^2$.
Similarly, if $f$ is merely sufficiently differentiable, its infinite expression in spherical harmonics converges (absolutely) uniformly pointwise, and can be squared as you indicated.
Details about how much differentiability, of various sorts, suffices are more complicated.
EDIT: For references for rigorous treatment: Stein and Weiss' old book "Introduction Fourier Analysis on Euclidean Spaces" has a chapter that covers many set-up issues about rigorous treatment of spherical harmonics, including things like estimate of sup norms in terms of $L^2$ norms. Also, Wikipedia's page on "spherical harmonics" does include several remarks about convergence arguments. My own course notes from introductory modular forms courses (at http://www.math.umn.edu/~garrett/m/mfms/) include some essays that treat the $L^2$ Sobolev theory, in the context of Weyl's criterion for equidistribution on spheres: http://www.math.umn.edu/~garrett/m/mfms/notes_2014-15/09_spheres.pdf
One basic assertion about convergence via Sobolev theory is the following. For a test function $f$ on the sphere, and non-negative integer $k$, $|f|^2_k=langle (1-Delta)^k f,frangle$ is the $k$th Sobolev norm (squared). Let $H^k$ be the completion of test functions with respect to that norm. Then a basic (not-so-trivial) theorem is that for $k>n/2$ the space $H^k$ lies inside the space of continuous functions, and the convergence of the finite partial sums of the spherical harmonic expansion converge uniformly pointwise to the function.
answered Jan 20 at 15:56
paul garrettpaul garrett
32.1k362120
$endgroup$
Yes, as you anticipate, although the spirit of "every function can be representated as an infinite sum of spherical harmonics", there are problems with convergence, as well as problems with the meaning of multiplication of two infinite sums of functions. These issues already arise (famously) on the one-sphere and the theory of Fourier series.
As a clear simple case: if the function $f$ is smooth, then the infinite sum of spherical harmonics converges pointwise uniformly to it, can be differentiated termwise, and all derivatives' infinite series expansions converge uniformly pointwise as well. And then it is immediate that the series can be squared and express $f^2$.
Similarly, if $f$ is merely sufficiently differentiable, its infinite expression in spherical harmonics converges (absolutely) uniformly pointwise, and can be squared as you indicated.
Details about how much differentiability, of various sorts, suffices are more complicated.
EDIT: For references for rigorous treatment: Stein and Weiss' old book "Introduction Fourier Analysis on Euclidean Spaces" has a chapter that covers many set-up issues about rigorous treatment of spherical harmonics, including things like estimate of sup norms in terms of $L^2$ norms. Also, Wikipedia's page on "spherical harmonics" does include several remarks about convergence arguments. My own course notes from introductory modular forms courses (at http://www.math.umn.edu/~garrett/m/mfms/) include some essays that treat the $L^2$ Sobolev theory, in the context of Weyl's criterion for equidistribution on spheres: http://www.math.umn.edu/~garrett/m/mfms/notes_2014-15/09_spheres.pdf
One basic assertion about convergence via Sobolev theory is the following. For a test function $f$ on the sphere, and non-negative integer $k$, $|f|^2_k=langle (1-Delta)^k f,frangle$ is the $k$th Sobolev norm (squared). Let $H^k$ be the completion of test functions with respect to that norm. Then a basic (not-so-trivial) theorem is that for $k>n/2$ the space $H^k$ lies inside the space of continuous functions, and the convergence of the finite partial sums of the spherical harmonic expansion converge uniformly pointwise to the function.
answered Jan 20 at 15:56
paul garrettpaul garrett
32.1k362120
edited Jan 21 at 16:48
answered Jan 20 at 15:56
paul garrettpaul garrett
32.1k362120
answered Jan 20 at 15:56
paul garrettpaul garrett
32.1k362120
answered Jan 20 at 15:56
paul garrettpaul garrett
32.1k362120
32.1k362120
$begingroup$
Thank you, Paul. May I ask you to elaborate. At least, some literature and where to start from. I am not familiar with the subject. Thank you.
$endgroup$
– user124297
Jan 21 at 12:20
add a comment |
$begingroup$
Thank you, Paul. May I ask you to elaborate. At least, some literature and where to start from. I am not familiar with the subject. Thank you.
$endgroup$
– user124297
Jan 21 at 12:20
$begingroup$
Thank you, Paul. May I ask you to elaborate. At least, some literature and where to start from. I am not familiar with the subject. Thank you.
$endgroup$
– user124297
Jan 21 at 12:20
$begingroup$
Thank you, Paul. May I ask you to elaborate. At least, some literature and where to start from. I am not familiar with the subject. Thank you.
$endgroup$
– user124297
Jan 21 at 12:20
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3075812%2fon-the-limit-of-the-finite-representation-of-harmonics%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
There is something unclear about your question: if $f$ itself is a finite linear combination of spherical harmonics, then that finite linear combination converges pointwise very well, and, of course, we can square both sides of the equality and rearrange. What limit do you mean to ask about? More general functions $f$? Then, yes, there will be non-trivial convergence issues, and non-trivial questions about squaring such an expansion. Can you clarify?
$endgroup$
– paul garrett
Jan 19 at 20:05
$begingroup$
@Paul Garrett: yes, I would like to consider more general function f, where f is any even function on the sphere. thank you.
$endgroup$
– user124297
Jan 20 at 15:19
$begingroup$
@Paul Garrett: there are two questions here. The main one-yes, I would like to consider any even function f on the sphere. Another question is whether there is a representation of giving f through infinite combination of spherical harmonics. Thank you.
$endgroup$
– user124297
Jan 20 at 15:21