Mixing
Could families of “Airys” and “Bairys” of integer “frequencies” be useful?
$begingroup$
A very famous family of functions are the complex exponentials and in the case of real valued functions, the sin and cos functions. They are related by the famous Euler formulas:
$$exp(iphi) = cos(phi)+isin(phi)$$
For example Discrete Fourier Transform with immense usefulness in science and engineering use basis functions like these:
$$sum_{n}^{}a_nexpleft[phi i nright]$$
or in the real case:
$$sum_{n}^{}a_ncos(phi n) + b_nsin(phi n)$$
Where integers $n$ decide how different frequencies these basis functions have.
Another interesting set of functions are the Airy and Bairy functions which are solutions to the following differential equation:
$$y''(x)-xy=0$$
but sadly don't have any simple analytic formula (as far as I know)
They are related in some respects like sin and cos are. By "ocular inspection" (and an engineering mindset) we see that they are "out of phase" with each other, always peaking at the other's zero.
Having observed the above properties, can we define some similar kind of linear transform in which we and up with basis functions consisting of $$Ai(nx), Bi(nx), ninleft{0,1,cdots,right}$$
pde fourier-analysis physics applications fast-fourier-transform
asked Jan 23 at 15:42
mathreadlermathreadler
15.1k72263
$endgroup$
add a comment |
$begingroup$
A very famous family of functions are the complex exponentials and in the case of real valued functions, the sin and cos functions. They are related by the famous Euler formulas:
$$exp(iphi) = cos(phi)+isin(phi)$$
For example Discrete Fourier Transform with immense usefulness in science and engineering use basis functions like these:
$$sum_{n}^{}a_nexpleft[phi i nright]$$
or in the real case:
$$sum_{n}^{}a_ncos(phi n) + b_nsin(phi n)$$
Where integers $n$ decide how different frequencies these basis functions have.
Another interesting set of functions are the Airy and Bairy functions which are solutions to the following differential equation:
$$y''(x)-xy=0$$
but sadly don't have any simple analytic formula (as far as I know)
They are related in some respects like sin and cos are. By "ocular inspection" (and an engineering mindset) we see that they are "out of phase" with each other, always peaking at the other's zero.
Having observed the above properties, can we define some similar kind of linear transform in which we and up with basis functions consisting of $$Ai(nx), Bi(nx), ninleft{0,1,cdots,right}$$
pde fourier-analysis physics applications fast-fourier-transform
asked Jan 23 at 15:42
mathreadlermathreadler
15.1k72263
$endgroup$
add a comment |
$begingroup$
A very famous family of functions are the complex exponentials and in the case of real valued functions, the sin and cos functions. They are related by the famous Euler formulas:
$$exp(iphi) = cos(phi)+isin(phi)$$
For example Discrete Fourier Transform with immense usefulness in science and engineering use basis functions like these:
$$sum_{n}^{}a_nexpleft[phi i nright]$$
or in the real case:
$$sum_{n}^{}a_ncos(phi n) + b_nsin(phi n)$$
Where integers $n$ decide how different frequencies these basis functions have.
Another interesting set of functions are the Airy and Bairy functions which are solutions to the following differential equation:
$$y''(x)-xy=0$$
but sadly don't have any simple analytic formula (as far as I know)
They are related in some respects like sin and cos are. By "ocular inspection" (and an engineering mindset) we see that they are "out of phase" with each other, always peaking at the other's zero.
Having observed the above properties, can we define some similar kind of linear transform in which we and up with basis functions consisting of $$Ai(nx), Bi(nx), ninleft{0,1,cdots,right}$$
pde fourier-analysis physics applications fast-fourier-transform
asked Jan 23 at 15:42
mathreadlermathreadler
15.1k72263
$endgroup$
A very famous family of functions are the complex exponentials and in the case of real valued functions, the sin and cos functions. They are related by the famous Euler formulas:
$$exp(iphi) = cos(phi)+isin(phi)$$
For example Discrete Fourier Transform with immense usefulness in science and engineering use basis functions like these:
$$sum_{n}^{}a_nexpleft[phi i nright]$$
or in the real case:
$$sum_{n}^{}a_ncos(phi n) + b_nsin(phi n)$$
Where integers $n$ decide how different frequencies these basis functions have.
Another interesting set of functions are the Airy and Bairy functions which are solutions to the following differential equation:
$$y''(x)-xy=0$$
but sadly don't have any simple analytic formula (as far as I know)
They are related in some respects like sin and cos are. By "ocular inspection" (and an engineering mindset) we see that they are "out of phase" with each other, always peaking at the other's zero.
Having observed the above properties, can we define some similar kind of linear transform in which we and up with basis functions consisting of $$Ai(nx), Bi(nx), ninleft{0,1,cdots,right}$$
pde fourier-analysis physics applications fast-fourier-transform
pde fourier-analysis physics applications fast-fourier-transform
asked Jan 23 at 15:42
mathreadlermathreadler
15.1k72263
asked Jan 23 at 15:42
mathreadlermathreadler
15.1k72263
edited Jan 24 at 15:12
mathreadler
asked Jan 23 at 15:42
mathreadlermathreadler
15.1k72263
asked Jan 23 at 15:42
mathreadlermathreadler
15.1k72263
asked Jan 23 at 15:42
mathreadlermathreadler
15.1k72263
15.1k72263
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Example: Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods.
Consider the Airy differential equation $u'' = lambda x u$ where $une 0$, $u(pm 1) = 0$ and $-1le xle 1$ ...
The $n$-th eigenfunction is given by $A_i(lambda_n^{1/3}x)$ where $A_i$ is the classic Airy function and $lambda_n$ is the eigenvalue...
answered Jan 24 at 19:00
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
34.6k42971
$endgroup$
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%2f3084640%2fcould-families-of-airys-and-bairys-of-integer-frequencies-be-useful%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
$begingroup$
Example: Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods.
Consider the Airy differential equation $u'' = lambda x u$ where $une 0$, $u(pm 1) = 0$ and $-1le xle 1$ ...
The $n$-th eigenfunction is given by $A_i(lambda_n^{1/3}x)$ where $A_i$ is the classic Airy function and $lambda_n$ is the eigenvalue...
answered Jan 24 at 19:00
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
34.6k42971
$endgroup$
add a comment |
$begingroup$
Example: Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods.
Consider the Airy differential equation $u'' = lambda x u$ where $une 0$, $u(pm 1) = 0$ and $-1le xle 1$ ...
The $n$-th eigenfunction is given by $A_i(lambda_n^{1/3}x)$ where $A_i$ is the classic Airy function and $lambda_n$ is the eigenvalue...
answered Jan 24 at 19:00
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
34.6k42971
$endgroup$
add a comment |
$begingroup$
Example: Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods.
Consider the Airy differential equation $u'' = lambda x u$ where $une 0$, $u(pm 1) = 0$ and $-1le xle 1$ ...
The $n$-th eigenfunction is given by $A_i(lambda_n^{1/3}x)$ where $A_i$ is the classic Airy function and $lambda_n$ is the eigenvalue...
answered Jan 24 at 19:00
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
34.6k42971
$endgroup$
Example: Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods.
Consider the Airy differential equation $u'' = lambda x u$ where $une 0$, $u(pm 1) = 0$ and $-1le xle 1$ ...
The $n$-th eigenfunction is given by $A_i(lambda_n^{1/3}x)$ where $A_i$ is the classic Airy function and $lambda_n$ is the eigenvalue...
answered Jan 24 at 19:00
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
34.6k42971
answered Jan 24 at 19:00
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
34.6k42971
answered Jan 24 at 19:00
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
34.6k42971
answered Jan 24 at 19:00
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
34.6k42971
34.6k42971
add a comment |
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%2f3084640%2fcould-families-of-airys-and-bairys-of-integer-frequencies-be-useful%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
