Mixing
Prove that $U=P*overline{P}$ iff $P$ is uniform (group representation exercise.)
$begingroup$
This exercise comes right after the introduction of Fourier Inversion Theorem and Plancherel’s Formula in Diaconis’s Group Representation in Probability and Statistics.
Let $P$ be a probability on $G$. Define $overline{P}(s)=P(s^{-1})$. Show that $U=P*overline{P}$ iff $P$ is uniform.
I just learned the concept and tried to pluck something straight to the formulas and it became even more baffling. How am I supposed to used them? There is a remark above the exercise that if $rho$ is unitary the Plancherel Formula can be written as
$$sum{f(s)h(s)}=frac{1}{|G|}sum{d_imathrm{Tr}(widehat{h}(rho_i)widehat{f}(rho_i)^{*})}$$
The hat signifies Fourier transformation and $*$ probably means conjugate transpose. I still don’t really know how the formula can be rewritten that way since I skip studying linear algebra and has only definition from Wikipedia regarding unitary matrices. I would be grateful for any help.
Note: The Plancherel Formula in the book is
$$sum{f(s^{-1})h(s)}=frac{1}{|G|}sum{d_imathrm{Tr}(widehat{h}(rho_i)widehat{f}(rho_i))}.$$
linear-algebra representation-theory
asked Jan 14 at 10:10
AhmbakAhmbak
390110
$endgroup$
add a comment |
$begingroup$
This exercise comes right after the introduction of Fourier Inversion Theorem and Plancherel’s Formula in Diaconis’s Group Representation in Probability and Statistics.
Let $P$ be a probability on $G$. Define $overline{P}(s)=P(s^{-1})$. Show that $U=P*overline{P}$ iff $P$ is uniform.
I just learned the concept and tried to pluck something straight to the formulas and it became even more baffling. How am I supposed to used them? There is a remark above the exercise that if $rho$ is unitary the Plancherel Formula can be written as
$$sum{f(s)h(s)}=frac{1}{|G|}sum{d_imathrm{Tr}(widehat{h}(rho_i)widehat{f}(rho_i)^{*})}$$
The hat signifies Fourier transformation and $*$ probably means conjugate transpose. I still don’t really know how the formula can be rewritten that way since I skip studying linear algebra and has only definition from Wikipedia regarding unitary matrices. I would be grateful for any help.
Note: The Plancherel Formula in the book is
$$sum{f(s^{-1})h(s)}=frac{1}{|G|}sum{d_imathrm{Tr}(widehat{h}(rho_i)widehat{f}(rho_i))}.$$
linear-algebra representation-theory
asked Jan 14 at 10:10
AhmbakAhmbak
390110
$endgroup$
add a comment |
$begingroup$
This exercise comes right after the introduction of Fourier Inversion Theorem and Plancherel’s Formula in Diaconis’s Group Representation in Probability and Statistics.
Let $P$ be a probability on $G$. Define $overline{P}(s)=P(s^{-1})$. Show that $U=P*overline{P}$ iff $P$ is uniform.
I just learned the concept and tried to pluck something straight to the formulas and it became even more baffling. How am I supposed to used them? There is a remark above the exercise that if $rho$ is unitary the Plancherel Formula can be written as
$$sum{f(s)h(s)}=frac{1}{|G|}sum{d_imathrm{Tr}(widehat{h}(rho_i)widehat{f}(rho_i)^{*})}$$
The hat signifies Fourier transformation and $*$ probably means conjugate transpose. I still don’t really know how the formula can be rewritten that way since I skip studying linear algebra and has only definition from Wikipedia regarding unitary matrices. I would be grateful for any help.
Note: The Plancherel Formula in the book is
$$sum{f(s^{-1})h(s)}=frac{1}{|G|}sum{d_imathrm{Tr}(widehat{h}(rho_i)widehat{f}(rho_i))}.$$
linear-algebra representation-theory
asked Jan 14 at 10:10
AhmbakAhmbak
390110
$endgroup$
This exercise comes right after the introduction of Fourier Inversion Theorem and Plancherel’s Formula in Diaconis’s Group Representation in Probability and Statistics.
Let $P$ be a probability on $G$. Define $overline{P}(s)=P(s^{-1})$. Show that $U=P*overline{P}$ iff $P$ is uniform.
I just learned the concept and tried to pluck something straight to the formulas and it became even more baffling. How am I supposed to used them? There is a remark above the exercise that if $rho$ is unitary the Plancherel Formula can be written as
$$sum{f(s)h(s)}=frac{1}{|G|}sum{d_imathrm{Tr}(widehat{h}(rho_i)widehat{f}(rho_i)^{*})}$$
The hat signifies Fourier transformation and $*$ probably means conjugate transpose. I still don’t really know how the formula can be rewritten that way since I skip studying linear algebra and has only definition from Wikipedia regarding unitary matrices. I would be grateful for any help.
Note: The Plancherel Formula in the book is
$$sum{f(s^{-1})h(s)}=frac{1}{|G|}sum{d_imathrm{Tr}(widehat{h}(rho_i)widehat{f}(rho_i))}.$$
linear-algebra representation-theory
linear-algebra representation-theory
asked Jan 14 at 10:10
AhmbakAhmbak
390110
asked Jan 14 at 10:10
AhmbakAhmbak
390110
asked Jan 14 at 10:10
AhmbakAhmbak
390110
asked Jan 14 at 10:10
AhmbakAhmbak
390110
asked Jan 14 at 10:10
AhmbakAhmbak
390110
390110
add a comment |
add a comment |
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