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Proving the Parseval’s identity via inner products
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I am proving the identity $langle f, grangle = langlehatf, hatgrangle$, using the Discrete Fourier analysis, which is the Parseval’s identity.
I already know that if $$langle f, grangle=sum_xin mathbbZ_nf(x)overlineg(x),$$
$$w_n = expleft(frac2Àixnright)$$ and $hatf$ is the fourier transform of $f$. Then
begineqnarray
% nonumber to remove numbering (before each equation)
langlehatf, hatgrangle&=&sum_xin mathbbZ_nhatf(x)overlinehatg(x) \
&=& sum_xinmathbbZ_nleft( left(sum_xinmathbbZ_nf(x)w^-rxright)overlineleft(sum_xinmathbbZ_ng(x)w^-sx right) right)\
&=&
endeqnarray
I don't know whether I am on the right track and what next follow?
functional-analysis pde fourier-analysis
edited Jul 15 at 13:30
mechanodroid
22.3k52041
asked Jul 15 at 13:08
Sulayman
20227
add a comment |Â
up vote
2
down vote
favorite
I am proving the identity $langle f, grangle = langlehatf, hatgrangle$, using the Discrete Fourier analysis, which is the Parseval’s identity.
I already know that if $$langle f, grangle=sum_xin mathbbZ_nf(x)overlineg(x),$$
$$w_n = expleft(frac2Àixnright)$$ and $hatf$ is the fourier transform of $f$. Then
begineqnarray
% nonumber to remove numbering (before each equation)
langlehatf, hatgrangle&=&sum_xin mathbbZ_nhatf(x)overlinehatg(x) \
&=& sum_xinmathbbZ_nleft( left(sum_xinmathbbZ_nf(x)w^-rxright)overlineleft(sum_xinmathbbZ_ng(x)w^-sx right) right)\
&=&
endeqnarray
I don't know whether I am on the right track and what next follow?
functional-analysis pde fourier-analysis
edited Jul 15 at 13:30
mechanodroid
22.3k52041
asked Jul 15 at 13:08
Sulayman
20227
Right track, but you need to differentiate the indices in your sums (for example, you might make them x, y, and z). Otherwise you will confuse yourself. Eventually you will be able to isolate a sum that leads to a delta function.
– John Polcari
Jul 15 at 13:18
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am proving the identity $langle f, grangle = langlehatf, hatgrangle$, using the Discrete Fourier analysis, which is the Parseval’s identity.
I already know that if $$langle f, grangle=sum_xin mathbbZ_nf(x)overlineg(x),$$
$$w_n = expleft(frac2Àixnright)$$ and $hatf$ is the fourier transform of $f$. Then
begineqnarray
% nonumber to remove numbering (before each equation)
langlehatf, hatgrangle&=&sum_xin mathbbZ_nhatf(x)overlinehatg(x) \
&=& sum_xinmathbbZ_nleft( left(sum_xinmathbbZ_nf(x)w^-rxright)overlineleft(sum_xinmathbbZ_ng(x)w^-sx right) right)\
&=&
endeqnarray
I don't know whether I am on the right track and what next follow?
functional-analysis pde fourier-analysis
edited Jul 15 at 13:30
mechanodroid
22.3k52041
asked Jul 15 at 13:08
Sulayman
20227
I am proving the identity $langle f, grangle = langlehatf, hatgrangle$, using the Discrete Fourier analysis, which is the Parseval’s identity.
I already know that if $$langle f, grangle=sum_xin mathbbZ_nf(x)overlineg(x),$$
$$w_n = expleft(frac2Àixnright)$$ and $hatf$ is the fourier transform of $f$. Then
begineqnarray
% nonumber to remove numbering (before each equation)
langlehatf, hatgrangle&=&sum_xin mathbbZ_nhatf(x)overlinehatg(x) \
&=& sum_xinmathbbZ_nleft( left(sum_xinmathbbZ_nf(x)w^-rxright)overlineleft(sum_xinmathbbZ_ng(x)w^-sx right) right)\
&=&
endeqnarray
I don't know whether I am on the right track and what next follow?
functional-analysis pde fourier-analysis
edited Jul 15 at 13:30
mechanodroid
22.3k52041
asked Jul 15 at 13:08
Sulayman
20227
edited Jul 15 at 13:30
mechanodroid
22.3k52041
edited Jul 15 at 13:30
mechanodroid
22.3k52041
edited Jul 15 at 13:30
mechanodroid
22.3k52041
22.3k52041
asked Jul 15 at 13:08
Sulayman
20227
asked Jul 15 at 13:08
Sulayman
20227
asked Jul 15 at 13:08
Sulayman
20227
20227
Right track, but you need to differentiate the indices in your sums (for example, you might make them x, y, and z). Otherwise you will confuse yourself. Eventually you will be able to isolate a sum that leads to a delta function.
– John Polcari
Jul 15 at 13:18
add a comment |Â
Right track, but you need to differentiate the indices in your sums (for example, you might make them x, y, and z). Otherwise you will confuse yourself. Eventually you will be able to isolate a sum that leads to a delta function.
– John Polcari
Jul 15 at 13:18
Right track, but you need to differentiate the indices in your sums (for example, you might make them x, y, and z). Otherwise you will confuse yourself. Eventually you will be able to isolate a sum that leads to a delta function.
– John Polcari
Jul 15 at 13:18
Right track, but you need to differentiate the indices in your sums (for example, you might make them x, y, and z). Otherwise you will confuse yourself. Eventually you will be able to isolate a sum that leads to a delta function.
– John Polcari
Jul 15 at 13:18
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
We know that $(w_n)_ninmathbbZ$ is an orthonormal basis so
$$langle f, grangle = leftlangle sum_m in mathbbZ hatf(m)w_m, sum_n in mathbbZ hatg(n)w_nrightrangle = sum_minmathbbZsum_ninmathbbZ hatf(m) overlinehatg(n) underbracelangle w_m, w_nrangle_=delta_mn = sum_n in mathbbZ hatf(n) overlinehatg(n) = langle hatf, hatgrangle$$
answered Jul 15 at 13:29
mechanodroid
22.3k52041
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
We know that $(w_n)_ninmathbbZ$ is an orthonormal basis so
$$langle f, grangle = leftlangle sum_m in mathbbZ hatf(m)w_m, sum_n in mathbbZ hatg(n)w_nrightrangle = sum_minmathbbZsum_ninmathbbZ hatf(m) overlinehatg(n) underbracelangle w_m, w_nrangle_=delta_mn = sum_n in mathbbZ hatf(n) overlinehatg(n) = langle hatf, hatgrangle$$
answered Jul 15 at 13:29
mechanodroid
22.3k52041
add a comment |Â
up vote
1
down vote
accepted
We know that $(w_n)_ninmathbbZ$ is an orthonormal basis so
$$langle f, grangle = leftlangle sum_m in mathbbZ hatf(m)w_m, sum_n in mathbbZ hatg(n)w_nrightrangle = sum_minmathbbZsum_ninmathbbZ hatf(m) overlinehatg(n) underbracelangle w_m, w_nrangle_=delta_mn = sum_n in mathbbZ hatf(n) overlinehatg(n) = langle hatf, hatgrangle$$
answered Jul 15 at 13:29
mechanodroid
22.3k52041
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
We know that $(w_n)_ninmathbbZ$ is an orthonormal basis so
$$langle f, grangle = leftlangle sum_m in mathbbZ hatf(m)w_m, sum_n in mathbbZ hatg(n)w_nrightrangle = sum_minmathbbZsum_ninmathbbZ hatf(m) overlinehatg(n) underbracelangle w_m, w_nrangle_=delta_mn = sum_n in mathbbZ hatf(n) overlinehatg(n) = langle hatf, hatgrangle$$
answered Jul 15 at 13:29
mechanodroid
22.3k52041
We know that $(w_n)_ninmathbbZ$ is an orthonormal basis so
$$langle f, grangle = leftlangle sum_m in mathbbZ hatf(m)w_m, sum_n in mathbbZ hatg(n)w_nrightrangle = sum_minmathbbZsum_ninmathbbZ hatf(m) overlinehatg(n) underbracelangle w_m, w_nrangle_=delta_mn = sum_n in mathbbZ hatf(n) overlinehatg(n) = langle hatf, hatgrangle$$
answered Jul 15 at 13:29
mechanodroid
22.3k52041
answered Jul 15 at 13:29
mechanodroid
22.3k52041
answered Jul 15 at 13:29
mechanodroid
22.3k52041
answered Jul 15 at 13:29
mechanodroid
22.3k52041
22.3k52041
add a comment |Â
add a comment |Â
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Right track, but you need to differentiate the indices in your sums (for example, you might make them x, y, and z). Otherwise you will confuse yourself. Eventually you will be able to isolate a sum that leads to a delta function.
– John Polcari
Jul 15 at 13:18