I am evaluating the fourier transform of a function + a constant:...












0












$begingroup$


I am evaluating the fourier transform of a function plus a constant $c$:



$$frac{1}{2pi}int_{-infty}^{+infty}(f(x)+c)e^{-ikx}dx.$$



As a result, I should get the fourier transform of the function $f(x)$ plus a dirac delta:
$$frac{1}{2pi}int_{-infty}^{+infty}(f(x)+c)e^{-ikx}dx=hat{f}(k)+cdelta(k).$$



However, I cannot understand whether I am making some mistake or not and the meaning of such a result.










share|cite|improve this question











$endgroup$












  • $begingroup$
    the integral $int_{-infty}^infty ce^{-ikt}, dt$ doesnt exists. And the Dirac delta is not a function, so Im not sure what you mean by $delta(k)$
    $endgroup$
    – Masacroso
    Jan 8 at 12:11












  • $begingroup$
    Your result is almost correct, the Fourier transform of $1$ is $delta$ with your normalization, so you should have $c delta(k)$.
    $endgroup$
    – Botond
    Jan 8 at 12:15












  • $begingroup$
    @Botond I cannot understand the physical meaning. The Fourier transform can be associated to an energy spectrum. In this case do I have an infinite energy at zero frequency?
    $endgroup$
    – ARF
    Jan 8 at 13:13










  • $begingroup$
    I can't really say anything about it without context. But why don't you ask it on PSE?
    $endgroup$
    – Botond
    Jan 8 at 13:35










  • $begingroup$
    I can only assume you are talking about the physical meaning as I do not see a question otherwise. In that case superimposing $cos(omega_0 t)$ with low $omega_0$ to a signal $f(t)$ will result in a disturbed signal on top. In the frequency domain you will see spikes at $pm omega_0$. Think of a low constant frequency bass sound on top of a melody. Now lot $omega_0 rightarrow 0$.
    $endgroup$
    – Diger
    Jan 8 at 14:32


















0












$begingroup$


I am evaluating the fourier transform of a function plus a constant $c$:



$$frac{1}{2pi}int_{-infty}^{+infty}(f(x)+c)e^{-ikx}dx.$$



As a result, I should get the fourier transform of the function $f(x)$ plus a dirac delta:
$$frac{1}{2pi}int_{-infty}^{+infty}(f(x)+c)e^{-ikx}dx=hat{f}(k)+cdelta(k).$$



However, I cannot understand whether I am making some mistake or not and the meaning of such a result.










share|cite|improve this question











$endgroup$












  • $begingroup$
    the integral $int_{-infty}^infty ce^{-ikt}, dt$ doesnt exists. And the Dirac delta is not a function, so Im not sure what you mean by $delta(k)$
    $endgroup$
    – Masacroso
    Jan 8 at 12:11












  • $begingroup$
    Your result is almost correct, the Fourier transform of $1$ is $delta$ with your normalization, so you should have $c delta(k)$.
    $endgroup$
    – Botond
    Jan 8 at 12:15












  • $begingroup$
    @Botond I cannot understand the physical meaning. The Fourier transform can be associated to an energy spectrum. In this case do I have an infinite energy at zero frequency?
    $endgroup$
    – ARF
    Jan 8 at 13:13










  • $begingroup$
    I can't really say anything about it without context. But why don't you ask it on PSE?
    $endgroup$
    – Botond
    Jan 8 at 13:35










  • $begingroup$
    I can only assume you are talking about the physical meaning as I do not see a question otherwise. In that case superimposing $cos(omega_0 t)$ with low $omega_0$ to a signal $f(t)$ will result in a disturbed signal on top. In the frequency domain you will see spikes at $pm omega_0$. Think of a low constant frequency bass sound on top of a melody. Now lot $omega_0 rightarrow 0$.
    $endgroup$
    – Diger
    Jan 8 at 14:32
















0












0








0





$begingroup$


I am evaluating the fourier transform of a function plus a constant $c$:



$$frac{1}{2pi}int_{-infty}^{+infty}(f(x)+c)e^{-ikx}dx.$$



As a result, I should get the fourier transform of the function $f(x)$ plus a dirac delta:
$$frac{1}{2pi}int_{-infty}^{+infty}(f(x)+c)e^{-ikx}dx=hat{f}(k)+cdelta(k).$$



However, I cannot understand whether I am making some mistake or not and the meaning of such a result.










share|cite|improve this question











$endgroup$




I am evaluating the fourier transform of a function plus a constant $c$:



$$frac{1}{2pi}int_{-infty}^{+infty}(f(x)+c)e^{-ikx}dx.$$



As a result, I should get the fourier transform of the function $f(x)$ plus a dirac delta:
$$frac{1}{2pi}int_{-infty}^{+infty}(f(x)+c)e^{-ikx}dx=hat{f}(k)+cdelta(k).$$



However, I cannot understand whether I am making some mistake or not and the meaning of such a result.







fourier-transform distribution-theory dirac-delta






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 13 at 23:19









Qmechanic

4,96711855




4,96711855










asked Jan 8 at 12:07









ARFARF

31




31












  • $begingroup$
    the integral $int_{-infty}^infty ce^{-ikt}, dt$ doesnt exists. And the Dirac delta is not a function, so Im not sure what you mean by $delta(k)$
    $endgroup$
    – Masacroso
    Jan 8 at 12:11












  • $begingroup$
    Your result is almost correct, the Fourier transform of $1$ is $delta$ with your normalization, so you should have $c delta(k)$.
    $endgroup$
    – Botond
    Jan 8 at 12:15












  • $begingroup$
    @Botond I cannot understand the physical meaning. The Fourier transform can be associated to an energy spectrum. In this case do I have an infinite energy at zero frequency?
    $endgroup$
    – ARF
    Jan 8 at 13:13










  • $begingroup$
    I can't really say anything about it without context. But why don't you ask it on PSE?
    $endgroup$
    – Botond
    Jan 8 at 13:35










  • $begingroup$
    I can only assume you are talking about the physical meaning as I do not see a question otherwise. In that case superimposing $cos(omega_0 t)$ with low $omega_0$ to a signal $f(t)$ will result in a disturbed signal on top. In the frequency domain you will see spikes at $pm omega_0$. Think of a low constant frequency bass sound on top of a melody. Now lot $omega_0 rightarrow 0$.
    $endgroup$
    – Diger
    Jan 8 at 14:32




















  • $begingroup$
    the integral $int_{-infty}^infty ce^{-ikt}, dt$ doesnt exists. And the Dirac delta is not a function, so Im not sure what you mean by $delta(k)$
    $endgroup$
    – Masacroso
    Jan 8 at 12:11












  • $begingroup$
    Your result is almost correct, the Fourier transform of $1$ is $delta$ with your normalization, so you should have $c delta(k)$.
    $endgroup$
    – Botond
    Jan 8 at 12:15












  • $begingroup$
    @Botond I cannot understand the physical meaning. The Fourier transform can be associated to an energy spectrum. In this case do I have an infinite energy at zero frequency?
    $endgroup$
    – ARF
    Jan 8 at 13:13










  • $begingroup$
    I can't really say anything about it without context. But why don't you ask it on PSE?
    $endgroup$
    – Botond
    Jan 8 at 13:35










  • $begingroup$
    I can only assume you are talking about the physical meaning as I do not see a question otherwise. In that case superimposing $cos(omega_0 t)$ with low $omega_0$ to a signal $f(t)$ will result in a disturbed signal on top. In the frequency domain you will see spikes at $pm omega_0$. Think of a low constant frequency bass sound on top of a melody. Now lot $omega_0 rightarrow 0$.
    $endgroup$
    – Diger
    Jan 8 at 14:32


















$begingroup$
the integral $int_{-infty}^infty ce^{-ikt}, dt$ doesnt exists. And the Dirac delta is not a function, so Im not sure what you mean by $delta(k)$
$endgroup$
– Masacroso
Jan 8 at 12:11






$begingroup$
the integral $int_{-infty}^infty ce^{-ikt}, dt$ doesnt exists. And the Dirac delta is not a function, so Im not sure what you mean by $delta(k)$
$endgroup$
– Masacroso
Jan 8 at 12:11














$begingroup$
Your result is almost correct, the Fourier transform of $1$ is $delta$ with your normalization, so you should have $c delta(k)$.
$endgroup$
– Botond
Jan 8 at 12:15






$begingroup$
Your result is almost correct, the Fourier transform of $1$ is $delta$ with your normalization, so you should have $c delta(k)$.
$endgroup$
– Botond
Jan 8 at 12:15














$begingroup$
@Botond I cannot understand the physical meaning. The Fourier transform can be associated to an energy spectrum. In this case do I have an infinite energy at zero frequency?
$endgroup$
– ARF
Jan 8 at 13:13




$begingroup$
@Botond I cannot understand the physical meaning. The Fourier transform can be associated to an energy spectrum. In this case do I have an infinite energy at zero frequency?
$endgroup$
– ARF
Jan 8 at 13:13












$begingroup$
I can't really say anything about it without context. But why don't you ask it on PSE?
$endgroup$
– Botond
Jan 8 at 13:35




$begingroup$
I can't really say anything about it without context. But why don't you ask it on PSE?
$endgroup$
– Botond
Jan 8 at 13:35












$begingroup$
I can only assume you are talking about the physical meaning as I do not see a question otherwise. In that case superimposing $cos(omega_0 t)$ with low $omega_0$ to a signal $f(t)$ will result in a disturbed signal on top. In the frequency domain you will see spikes at $pm omega_0$. Think of a low constant frequency bass sound on top of a melody. Now lot $omega_0 rightarrow 0$.
$endgroup$
– Diger
Jan 8 at 14:32






$begingroup$
I can only assume you are talking about the physical meaning as I do not see a question otherwise. In that case superimposing $cos(omega_0 t)$ with low $omega_0$ to a signal $f(t)$ will result in a disturbed signal on top. In the frequency domain you will see spikes at $pm omega_0$. Think of a low constant frequency bass sound on top of a melody. Now lot $omega_0 rightarrow 0$.
$endgroup$
– Diger
Jan 8 at 14:32












0






active

oldest

votes











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066096%2fi-am-evaluating-the-fourier-transform-of-a-function-a-constant-frac12-pi%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























0






active

oldest

votes








0






active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066096%2fi-am-evaluating-the-fourier-transform-of-a-function-a-constant-frac12-pi%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

1300-talet

1300-talet

Display a custom attribute below product name in the front-end Magento 1.9.3.8