Is there an elementary proof of Fourier's Theorem?












1












$begingroup$


Fourier's Theorem



An arbitrary periodic function $F(t)$ with period $T$ can be decomposed into a linear combination of the functions $f_n(t)$ and $g_n(t)$ where,
$$f_n(t)=sin frac {2πnt}{T}$$
$$g_n(t)=cos frac {2πnt}{T}$$
Mathematically,
$$F(t)=b_0 + b_1g_1(t) +b_2g_2(t) + …… + a_1f_1(t) +a_2f_2(t) + ……,$$
where $n$ is a non-negative integer and all of $a_i$,$b_i$ are real.



Problem



As $F(t)$ is periodic with period $T$ ,
$$F(t+nT)=F(t)…….… (1)$$.
What I want is that I can derive the RHS directly from equation (1). However, I am unable to do so, perhaps in lack of theoretical knowledge. Any suggestions are welcome.



Note



What ever proofs I have got as yet, they assume the decomposition for some complex exponential and justify it using some arguments. I don't want anything like that






complex-analysis fourier-analysis alternative-proof







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    It is not true that any periodic function can be written in the way you claim. At the very least, the function needs to be measurable. Also, it is not true that $int_0^T f(t) , dt =0$ if $f$ is T-periodic.
    $endgroup$
    – PhoemueX
    Dec 20 '18 at 11:39










  • $begingroup$
    @PhoemueX, sorry , I was quite biased to write the integral. As for now , I have correct it.
    $endgroup$
    – Awe Kumar Jha
    Dec 20 '18 at 11:48






  • 1




    $begingroup$
    Do you know the Dirichlet kernel ? The Fourier series is $lim_{N to infty} F ast D_N$ where $ast$ is the (periodic) convolution. Then we show that $D_N(t) approx N h(Nt)$ and under some conditions $lim_{N to infty} F ast D_N =lim_{N to infty} F ast N h(Nt) = F$. The alternative is to show that ${ e^{2i pi n t}}$ is an orthonormal family of $L^2([0,1])$ and that it is dense so it is an orthonormal basis and $F = sum_n langle F,e^{2i pi n t}rangle e^{2i pi n t}$
    $endgroup$
    – reuns
    Dec 23 '18 at 9:27












  • $begingroup$
    @reuns, where does the Dirichlet kernel itself come from? In case you can answer this very question , the proof is complete.
    $endgroup$
    – Awe Kumar Jha
    Dec 23 '18 at 9:44
















1












$begingroup$


Fourier's Theorem



An arbitrary periodic function $F(t)$ with period $T$ can be decomposed into a linear combination of the functions $f_n(t)$ and $g_n(t)$ where,
$$f_n(t)=sin frac {2πnt}{T}$$
$$g_n(t)=cos frac {2πnt}{T}$$
Mathematically,
$$F(t)=b_0 + b_1g_1(t) +b_2g_2(t) + …… + a_1f_1(t) +a_2f_2(t) + ……,$$
where $n$ is a non-negative integer and all of $a_i$,$b_i$ are real.



Problem



As $F(t)$ is periodic with period $T$ ,
$$F(t+nT)=F(t)…….… (1)$$.
What I want is that I can derive the RHS directly from equation (1). However, I am unable to do so, perhaps in lack of theoretical knowledge. Any suggestions are welcome.



Note



What ever proofs I have got as yet, they assume the decomposition for some complex exponential and justify it using some arguments. I don't want anything like that






complex-analysis fourier-analysis alternative-proof







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    It is not true that any periodic function can be written in the way you claim. At the very least, the function needs to be measurable. Also, it is not true that $int_0^T f(t) , dt =0$ if $f$ is T-periodic.
    $endgroup$
    – PhoemueX
    Dec 20 '18 at 11:39










  • $begingroup$
    @PhoemueX, sorry , I was quite biased to write the integral. As for now , I have correct it.
    $endgroup$
    – Awe Kumar Jha
    Dec 20 '18 at 11:48






  • 1




    $begingroup$
    Do you know the Dirichlet kernel ? The Fourier series is $lim_{N to infty} F ast D_N$ where $ast$ is the (periodic) convolution. Then we show that $D_N(t) approx N h(Nt)$ and under some conditions $lim_{N to infty} F ast D_N =lim_{N to infty} F ast N h(Nt) = F$. The alternative is to show that ${ e^{2i pi n t}}$ is an orthonormal family of $L^2([0,1])$ and that it is dense so it is an orthonormal basis and $F = sum_n langle F,e^{2i pi n t}rangle e^{2i pi n t}$
    $endgroup$
    – reuns
    Dec 23 '18 at 9:27












  • $begingroup$
    @reuns, where does the Dirichlet kernel itself come from? In case you can answer this very question , the proof is complete.
    $endgroup$
    – Awe Kumar Jha
    Dec 23 '18 at 9:44














1












1








1


1



$begingroup$


Fourier's Theorem



An arbitrary periodic function $F(t)$ with period $T$ can be decomposed into a linear combination of the functions $f_n(t)$ and $g_n(t)$ where,
$$f_n(t)=sin frac {2πnt}{T}$$
$$g_n(t)=cos frac {2πnt}{T}$$
Mathematically,
$$F(t)=b_0 + b_1g_1(t) +b_2g_2(t) + …… + a_1f_1(t) +a_2f_2(t) + ……,$$
where $n$ is a non-negative integer and all of $a_i$,$b_i$ are real.



Problem



As $F(t)$ is periodic with period $T$ ,
$$F(t+nT)=F(t)…….… (1)$$.
What I want is that I can derive the RHS directly from equation (1). However, I am unable to do so, perhaps in lack of theoretical knowledge. Any suggestions are welcome.



Note



What ever proofs I have got as yet, they assume the decomposition for some complex exponential and justify it using some arguments. I don't want anything like that






complex-analysis fourier-analysis alternative-proof







share|cite|improve this question











$endgroup$




Fourier's Theorem



An arbitrary periodic function $F(t)$ with period $T$ can be decomposed into a linear combination of the functions $f_n(t)$ and $g_n(t)$ where,
$$f_n(t)=sin frac {2πnt}{T}$$
$$g_n(t)=cos frac {2πnt}{T}$$
Mathematically,
$$F(t)=b_0 + b_1g_1(t) +b_2g_2(t) + …… + a_1f_1(t) +a_2f_2(t) + ……,$$
where $n$ is a non-negative integer and all of $a_i$,$b_i$ are real.



Problem



As $F(t)$ is periodic with period $T$ ,
$$F(t+nT)=F(t)…….… (1)$$.
What I want is that I can derive the RHS directly from equation (1). However, I am unable to do so, perhaps in lack of theoretical knowledge. Any suggestions are welcome.



Note



What ever proofs I have got as yet, they assume the decomposition for some complex exponential and justify it using some arguments. I don't want anything like that






complex-analysis fourier-analysis alternative-proof




complex-analysis fourier-analysis alternative-proof






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 23 '18 at 9:42







Awe Kumar Jha

















asked Dec 20 '18 at 11:30









Awe Kumar JhaAwe Kumar Jha

570113




570113








  • 4




    $begingroup$
    It is not true that any periodic function can be written in the way you claim. At the very least, the function needs to be measurable. Also, it is not true that $int_0^T f(t) , dt =0$ if $f$ is T-periodic.
    $endgroup$
    – PhoemueX
    Dec 20 '18 at 11:39










  • $begingroup$
    @PhoemueX, sorry , I was quite biased to write the integral. As for now , I have correct it.
    $endgroup$
    – Awe Kumar Jha
    Dec 20 '18 at 11:48






  • 1




    $begingroup$
    Do you know the Dirichlet kernel ? The Fourier series is $lim_{N to infty} F ast D_N$ where $ast$ is the (periodic) convolution. Then we show that $D_N(t) approx N h(Nt)$ and under some conditions $lim_{N to infty} F ast D_N =lim_{N to infty} F ast N h(Nt) = F$. The alternative is to show that ${ e^{2i pi n t}}$ is an orthonormal family of $L^2([0,1])$ and that it is dense so it is an orthonormal basis and $F = sum_n langle F,e^{2i pi n t}rangle e^{2i pi n t}$
    $endgroup$
    – reuns
    Dec 23 '18 at 9:27












  • $begingroup$
    @reuns, where does the Dirichlet kernel itself come from? In case you can answer this very question , the proof is complete.
    $endgroup$
    – Awe Kumar Jha
    Dec 23 '18 at 9:44














  • 4




    $begingroup$
    It is not true that any periodic function can be written in the way you claim. At the very least, the function needs to be measurable. Also, it is not true that $int_0^T f(t) , dt =0$ if $f$ is T-periodic.
    $endgroup$
    – PhoemueX
    Dec 20 '18 at 11:39










  • $begingroup$
    @PhoemueX, sorry , I was quite biased to write the integral. As for now , I have correct it.
    $endgroup$
    – Awe Kumar Jha
    Dec 20 '18 at 11:48






  • 1




    $begingroup$
    Do you know the Dirichlet kernel ? The Fourier series is $lim_{N to infty} F ast D_N$ where $ast$ is the (periodic) convolution. Then we show that $D_N(t) approx N h(Nt)$ and under some conditions $lim_{N to infty} F ast D_N =lim_{N to infty} F ast N h(Nt) = F$. The alternative is to show that ${ e^{2i pi n t}}$ is an orthonormal family of $L^2([0,1])$ and that it is dense so it is an orthonormal basis and $F = sum_n langle F,e^{2i pi n t}rangle e^{2i pi n t}$
    $endgroup$
    – reuns
    Dec 23 '18 at 9:27












  • $begingroup$
    @reuns, where does the Dirichlet kernel itself come from? In case you can answer this very question , the proof is complete.
    $endgroup$
    – Awe Kumar Jha
    Dec 23 '18 at 9:44








4




4




$begingroup$
It is not true that any periodic function can be written in the way you claim. At the very least, the function needs to be measurable. Also, it is not true that $int_0^T f(t) , dt =0$ if $f$ is T-periodic.
$endgroup$
– PhoemueX
Dec 20 '18 at 11:39




$begingroup$
It is not true that any periodic function can be written in the way you claim. At the very least, the function needs to be measurable. Also, it is not true that $int_0^T f(t) , dt =0$ if $f$ is T-periodic.
$endgroup$
– PhoemueX
Dec 20 '18 at 11:39












$begingroup$
@PhoemueX, sorry , I was quite biased to write the integral. As for now , I have correct it.
$endgroup$
– Awe Kumar Jha
Dec 20 '18 at 11:48




$begingroup$
@PhoemueX, sorry , I was quite biased to write the integral. As for now , I have correct it.
$endgroup$
– Awe Kumar Jha
Dec 20 '18 at 11:48




1




1




$begingroup$
Do you know the Dirichlet kernel ? The Fourier series is $lim_{N to infty} F ast D_N$ where $ast$ is the (periodic) convolution. Then we show that $D_N(t) approx N h(Nt)$ and under some conditions $lim_{N to infty} F ast D_N =lim_{N to infty} F ast N h(Nt) = F$. The alternative is to show that ${ e^{2i pi n t}}$ is an orthonormal family of $L^2([0,1])$ and that it is dense so it is an orthonormal basis and $F = sum_n langle F,e^{2i pi n t}rangle e^{2i pi n t}$
$endgroup$
– reuns
Dec 23 '18 at 9:27






$begingroup$
Do you know the Dirichlet kernel ? The Fourier series is $lim_{N to infty} F ast D_N$ where $ast$ is the (periodic) convolution. Then we show that $D_N(t) approx N h(Nt)$ and under some conditions $lim_{N to infty} F ast D_N =lim_{N to infty} F ast N h(Nt) = F$. The alternative is to show that ${ e^{2i pi n t}}$ is an orthonormal family of $L^2([0,1])$ and that it is dense so it is an orthonormal basis and $F = sum_n langle F,e^{2i pi n t}rangle e^{2i pi n t}$
$endgroup$
– reuns
Dec 23 '18 at 9:27














$begingroup$
@reuns, where does the Dirichlet kernel itself come from? In case you can answer this very question , the proof is complete.
$endgroup$
– Awe Kumar Jha
Dec 23 '18 at 9:44




$begingroup$
@reuns, where does the Dirichlet kernel itself come from? In case you can answer this very question , the proof is complete.
$endgroup$
– Awe Kumar Jha
Dec 23 '18 at 9:44










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%2f3047443%2fis-there-an-elementary-proof-of-fouriers-theorem%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%2f3047443%2fis-there-an-elementary-proof-of-fouriers-theorem%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

Bundesstraße 106

Image
Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück ...
Read more

Verónica Boquete

Image
Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl ...
Read more

Ida-Boy-Ed-Garten

Image
Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich ...
Read more