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

43813




43813








  • 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










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