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Filling gaps in “a proof†of Fourier inversion formula
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Suppose $fin L^1(mathbbR)$. Define the Fourier transform of $f$ as:
$$hatf:mathbbRrightarrowmathbbC, ximapstoint_mathbbRf(t)e^-2pi ixi toperatornamedt.$$
Suppose that $hat fin L^1(mathbbR)$.
Under what further hypothesis on $f$ and $hatf$can we make rigorous the following argument:
$$forall xinmathbbR, int_mathbbR hat f(xi)e^2pi ixi xoperatornamedxi overset(1)= lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(hatf(nDelta L)e^2pi Delta L in xDelta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(left(int_mathbbRf(t)e^-2piDelta L i n toperatornamedtright)e^2pi in Delta LxDelta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(int_mathbbRf(t)e^2piDelta L i n (x-t)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(sum_k=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n (x-t)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(sum_k=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n left((x+frackDelta L)-tright)operatornamedt Delta Lright) \ overset(2)= lim_Delta Lrightarrow0^+ sum_k=-infty^+infty left(sum_n=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n left((x+frackDelta L)-tright)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_k=-infty^+infty left(sum_n=-infty^+infty left(frac1frac1Delta Lint_frac1Delta L[k,k+1]f(t)e^-2piDelta L i n toperatornamedtright)e^2pi Delta L in(x+frackDelta L)right) \ overset(star)= lim_Delta Lrightarrow0^+ sum_k=-infty^+infty fleft(x+frackDelta Lright)overset(3)=f(x) ?$$
The idea is to prove Fourier inversion formula in $mathbbR$ using the Fourier inversion formula on the torus, that we used in $(star)$.
In particular, while it seems that we can take care of $(1)$ and $(3)$ imposing some decreasing condition at $infty$ on $f$ and $hatf$, I'm having hard times in finding hypothesis on $f$ and $hatf$ for which $(2)$ holds, neither I have an idea on what technique I could use to prove it.
real-analysis fourier-analysis fourier-series fourier-transform
asked Jul 29 at 22:55
Bob
1,517522
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Suppose $fin L^1(mathbbR)$. Define the Fourier transform of $f$ as:
$$hatf:mathbbRrightarrowmathbbC, ximapstoint_mathbbRf(t)e^-2pi ixi toperatornamedt.$$
Suppose that $hat fin L^1(mathbbR)$.
Under what further hypothesis on $f$ and $hatf$can we make rigorous the following argument:
$$forall xinmathbbR, int_mathbbR hat f(xi)e^2pi ixi xoperatornamedxi overset(1)= lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(hatf(nDelta L)e^2pi Delta L in xDelta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(left(int_mathbbRf(t)e^-2piDelta L i n toperatornamedtright)e^2pi in Delta LxDelta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(int_mathbbRf(t)e^2piDelta L i n (x-t)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(sum_k=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n (x-t)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(sum_k=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n left((x+frackDelta L)-tright)operatornamedt Delta Lright) \ overset(2)= lim_Delta Lrightarrow0^+ sum_k=-infty^+infty left(sum_n=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n left((x+frackDelta L)-tright)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_k=-infty^+infty left(sum_n=-infty^+infty left(frac1frac1Delta Lint_frac1Delta L[k,k+1]f(t)e^-2piDelta L i n toperatornamedtright)e^2pi Delta L in(x+frackDelta L)right) \ overset(star)= lim_Delta Lrightarrow0^+ sum_k=-infty^+infty fleft(x+frackDelta Lright)overset(3)=f(x) ?$$
The idea is to prove Fourier inversion formula in $mathbbR$ using the Fourier inversion formula on the torus, that we used in $(star)$.
In particular, while it seems that we can take care of $(1)$ and $(3)$ imposing some decreasing condition at $infty$ on $f$ and $hatf$, I'm having hard times in finding hypothesis on $f$ and $hatf$ for which $(2)$ holds, neither I have an idea on what technique I could use to prove it.
real-analysis fourier-analysis fourier-series fourier-transform
asked Jul 29 at 22:55
Bob
1,517522
An application of Fubini's theorem
– timur
Aug 2 at 1:31
@timur yes, and how do you prove that the hypothesis of Fubini are satisfied?
– Bob
Aug 2 at 1:34
I know, if I had an answer it would not have been a comment :)
– timur
Aug 2 at 1:50
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
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Suppose $fin L^1(mathbbR)$. Define the Fourier transform of $f$ as:
$$hatf:mathbbRrightarrowmathbbC, ximapstoint_mathbbRf(t)e^-2pi ixi toperatornamedt.$$
Suppose that $hat fin L^1(mathbbR)$.
Under what further hypothesis on $f$ and $hatf$can we make rigorous the following argument:
$$forall xinmathbbR, int_mathbbR hat f(xi)e^2pi ixi xoperatornamedxi overset(1)= lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(hatf(nDelta L)e^2pi Delta L in xDelta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(left(int_mathbbRf(t)e^-2piDelta L i n toperatornamedtright)e^2pi in Delta LxDelta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(int_mathbbRf(t)e^2piDelta L i n (x-t)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(sum_k=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n (x-t)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(sum_k=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n left((x+frackDelta L)-tright)operatornamedt Delta Lright) \ overset(2)= lim_Delta Lrightarrow0^+ sum_k=-infty^+infty left(sum_n=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n left((x+frackDelta L)-tright)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_k=-infty^+infty left(sum_n=-infty^+infty left(frac1frac1Delta Lint_frac1Delta L[k,k+1]f(t)e^-2piDelta L i n toperatornamedtright)e^2pi Delta L in(x+frackDelta L)right) \ overset(star)= lim_Delta Lrightarrow0^+ sum_k=-infty^+infty fleft(x+frackDelta Lright)overset(3)=f(x) ?$$
The idea is to prove Fourier inversion formula in $mathbbR$ using the Fourier inversion formula on the torus, that we used in $(star)$.
In particular, while it seems that we can take care of $(1)$ and $(3)$ imposing some decreasing condition at $infty$ on $f$ and $hatf$, I'm having hard times in finding hypothesis on $f$ and $hatf$ for which $(2)$ holds, neither I have an idea on what technique I could use to prove it.
real-analysis fourier-analysis fourier-series fourier-transform
asked Jul 29 at 22:55
Bob
1,517522
Suppose $fin L^1(mathbbR)$. Define the Fourier transform of $f$ as:
$$hatf:mathbbRrightarrowmathbbC, ximapstoint_mathbbRf(t)e^-2pi ixi toperatornamedt.$$
Suppose that $hat fin L^1(mathbbR)$.
Under what further hypothesis on $f$ and $hatf$can we make rigorous the following argument:
$$forall xinmathbbR, int_mathbbR hat f(xi)e^2pi ixi xoperatornamedxi overset(1)= lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(hatf(nDelta L)e^2pi Delta L in xDelta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(left(int_mathbbRf(t)e^-2piDelta L i n toperatornamedtright)e^2pi in Delta LxDelta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(int_mathbbRf(t)e^2piDelta L i n (x-t)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(sum_k=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n (x-t)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_n=-infty^+infty left(sum_k=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n left((x+frackDelta L)-tright)operatornamedt Delta Lright) \ overset(2)= lim_Delta Lrightarrow0^+ sum_k=-infty^+infty left(sum_n=-infty^+inftyint_frac1Delta L[k,k+1]f(t)e^2piDelta L i n left((x+frackDelta L)-tright)operatornamedt Delta Lright) \ = lim_Delta Lrightarrow0^+ sum_k=-infty^+infty left(sum_n=-infty^+infty left(frac1frac1Delta Lint_frac1Delta L[k,k+1]f(t)e^-2piDelta L i n toperatornamedtright)e^2pi Delta L in(x+frackDelta L)right) \ overset(star)= lim_Delta Lrightarrow0^+ sum_k=-infty^+infty fleft(x+frackDelta Lright)overset(3)=f(x) ?$$
The idea is to prove Fourier inversion formula in $mathbbR$ using the Fourier inversion formula on the torus, that we used in $(star)$.
In particular, while it seems that we can take care of $(1)$ and $(3)$ imposing some decreasing condition at $infty$ on $f$ and $hatf$, I'm having hard times in finding hypothesis on $f$ and $hatf$ for which $(2)$ holds, neither I have an idea on what technique I could use to prove it.
real-analysis fourier-analysis fourier-series fourier-transform
asked Jul 29 at 22:55
Bob
1,517522
edited Jul 30 at 6:56
asked Jul 29 at 22:55
Bob
1,517522
asked Jul 29 at 22:55
Bob
1,517522
asked Jul 29 at 22:55
Bob
1,517522
1,517522
An application of Fubini's theorem
– timur
Aug 2 at 1:31
@timur yes, and how do you prove that the hypothesis of Fubini are satisfied?
– Bob
Aug 2 at 1:34
I know, if I had an answer it would not have been a comment :)
– timur
Aug 2 at 1:50
add a comment |Â
An application of Fubini's theorem
– timur
Aug 2 at 1:31
@timur yes, and how do you prove that the hypothesis of Fubini are satisfied?
– Bob
Aug 2 at 1:34
I know, if I had an answer it would not have been a comment :)
– timur
Aug 2 at 1:50
An application of Fubini's theorem
– timur
Aug 2 at 1:31
An application of Fubini's theorem
– timur
Aug 2 at 1:31
@timur yes, and how do you prove that the hypothesis of Fubini are satisfied?
– Bob
Aug 2 at 1:34
@timur yes, and how do you prove that the hypothesis of Fubini are satisfied?
– Bob
Aug 2 at 1:34
I know, if I had an answer it would not have been a comment :)
– timur
Aug 2 at 1:50
I know, if I had an answer it would not have been a comment :)
– timur
Aug 2 at 1:50
add a comment |Â
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An application of Fubini's theorem
– timur
Aug 2 at 1:31
@timur yes, and how do you prove that the hypothesis of Fubini are satisfied?
– Bob
Aug 2 at 1:34
I know, if I had an answer it would not have been a comment :)
– timur
Aug 2 at 1:50