Is there an elementary proof of Fourier's Theorem?
$begingroup$
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
asked Dec 20 '18 at 11:30
Awe Kumar JhaAwe Kumar Jha
43813
$endgroup$
add a comment |
$begingroup$
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
asked Dec 20 '18 at 11:30
Awe Kumar JhaAwe Kumar Jha
43813
$endgroup$
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
add a comment |
$begingroup$
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
asked Dec 20 '18 at 11:30
Awe Kumar JhaAwe Kumar Jha
43813
$endgroup$
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
asked Dec 20 '18 at 11:30
Awe Kumar JhaAwe Kumar Jha
43813
asked Dec 20 '18 at 11:30
Awe Kumar JhaAwe Kumar Jha
43813
edited Dec 23 '18 at 9:42
Awe Kumar Jha
asked Dec 20 '18 at 11:30
Awe Kumar JhaAwe Kumar Jha
43813
asked Dec 20 '18 at 11:30
Awe Kumar JhaAwe Kumar Jha
43813
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
add a comment |
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
add a comment |
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$begingroup$
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