How to solve equations of the form $fT=delta_0$?
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I want to solve distributional equations of the form $fT=delta_0$ for a $C^infty$ function $f$.
For the equation $fT=0$, we can bound the support of $T$ by
$$text{supp }Tsubset f^{-1}({0})$$
and that usually helps solving such equations.
However, for $fT=delta_0$ I don't seem to understand a pattern. For example, let's understand the solution of $xT=delta_0$:
We want to find a distribution $T$ such that $T(xvarphi)=varphi(0)$ for every $varphiinmathcal{D}(mathbb{R})$. Since we can't evaluate $varphi(x)/x$ at $0$, the next logical thing to do is to define $T$ as
$$T(varphi)=lim_{xto 0}frac{varphi(x)}{x}.$$
Since $varphi$ is $C^infty$, this works if and only if $varphi(0)=0$. We can fix this by defining
$$T(varphi)=lim_{xto 0}frac{varphi(x)-varphi(0)}{x}=varphi'(0).$$
That is, $-delta'_0$ is a particular solution of the equation $xT=delta_0$.
Now, notice that this method cant be generalized to solve equations of the form $fT=delta_0$ if $f$ is equal to $e^{-1/x}$ for $xgeq 0$ and $0$ otherwise because $f^{(k)}(0)=0$ for all $k$.
What should we do in this case? Is there a good method for solving general equations of the form $fT=delta_0$?
distribution-theory
asked Jan 4 at 9:33
Gabriel RibeiroGabriel Ribeiro
1,456523
$endgroup$
add a comment |
$begingroup$
I want to solve distributional equations of the form $fT=delta_0$ for a $C^infty$ function $f$.
For the equation $fT=0$, we can bound the support of $T$ by
$$text{supp }Tsubset f^{-1}({0})$$
and that usually helps solving such equations.
However, for $fT=delta_0$ I don't seem to understand a pattern. For example, let's understand the solution of $xT=delta_0$:
We want to find a distribution $T$ such that $T(xvarphi)=varphi(0)$ for every $varphiinmathcal{D}(mathbb{R})$. Since we can't evaluate $varphi(x)/x$ at $0$, the next logical thing to do is to define $T$ as
$$T(varphi)=lim_{xto 0}frac{varphi(x)}{x}.$$
Since $varphi$ is $C^infty$, this works if and only if $varphi(0)=0$. We can fix this by defining
$$T(varphi)=lim_{xto 0}frac{varphi(x)-varphi(0)}{x}=varphi'(0).$$
That is, $-delta'_0$ is a particular solution of the equation $xT=delta_0$.
Now, notice that this method cant be generalized to solve equations of the form $fT=delta_0$ if $f$ is equal to $e^{-1/x}$ for $xgeq 0$ and $0$ otherwise because $f^{(k)}(0)=0$ for all $k$.
What should we do in this case? Is there a good method for solving general equations of the form $fT=delta_0$?
distribution-theory
asked Jan 4 at 9:33
Gabriel RibeiroGabriel Ribeiro
1,456523
$endgroup$
$begingroup$
Short answer: no.
$endgroup$
– md2perpe
Jan 4 at 11:10
$begingroup$
@md2perpe The technique I used for solving equations of the form $fT=0$ was very useful to me. Are there other similar techniques?
$endgroup$
– Gabriel Ribeiro
Jan 4 at 11:46
$begingroup$
I know no technique to solve the equation when there is some point where $f^{(k)} = 0$ for all $k = 0, 1, 2, ldots$
$endgroup$
– md2perpe
Jan 4 at 14:54
add a comment |
$begingroup$
I want to solve distributional equations of the form $fT=delta_0$ for a $C^infty$ function $f$.
For the equation $fT=0$, we can bound the support of $T$ by
$$text{supp }Tsubset f^{-1}({0})$$
and that usually helps solving such equations.
However, for $fT=delta_0$ I don't seem to understand a pattern. For example, let's understand the solution of $xT=delta_0$:
We want to find a distribution $T$ such that $T(xvarphi)=varphi(0)$ for every $varphiinmathcal{D}(mathbb{R})$. Since we can't evaluate $varphi(x)/x$ at $0$, the next logical thing to do is to define $T$ as
$$T(varphi)=lim_{xto 0}frac{varphi(x)}{x}.$$
Since $varphi$ is $C^infty$, this works if and only if $varphi(0)=0$. We can fix this by defining
$$T(varphi)=lim_{xto 0}frac{varphi(x)-varphi(0)}{x}=varphi'(0).$$
That is, $-delta'_0$ is a particular solution of the equation $xT=delta_0$.
Now, notice that this method cant be generalized to solve equations of the form $fT=delta_0$ if $f$ is equal to $e^{-1/x}$ for $xgeq 0$ and $0$ otherwise because $f^{(k)}(0)=0$ for all $k$.
What should we do in this case? Is there a good method for solving general equations of the form $fT=delta_0$?
distribution-theory
asked Jan 4 at 9:33
Gabriel RibeiroGabriel Ribeiro
1,456523
$endgroup$
I want to solve distributional equations of the form $fT=delta_0$ for a $C^infty$ function $f$.
For the equation $fT=0$, we can bound the support of $T$ by
$$text{supp }Tsubset f^{-1}({0})$$
and that usually helps solving such equations.
However, for $fT=delta_0$ I don't seem to understand a pattern. For example, let's understand the solution of $xT=delta_0$:
We want to find a distribution $T$ such that $T(xvarphi)=varphi(0)$ for every $varphiinmathcal{D}(mathbb{R})$. Since we can't evaluate $varphi(x)/x$ at $0$, the next logical thing to do is to define $T$ as
$$T(varphi)=lim_{xto 0}frac{varphi(x)}{x}.$$
Since $varphi$ is $C^infty$, this works if and only if $varphi(0)=0$. We can fix this by defining
$$T(varphi)=lim_{xto 0}frac{varphi(x)-varphi(0)}{x}=varphi'(0).$$
That is, $-delta'_0$ is a particular solution of the equation $xT=delta_0$.
Now, notice that this method cant be generalized to solve equations of the form $fT=delta_0$ if $f$ is equal to $e^{-1/x}$ for $xgeq 0$ and $0$ otherwise because $f^{(k)}(0)=0$ for all $k$.
What should we do in this case? Is there a good method for solving general equations of the form $fT=delta_0$?
distribution-theory
distribution-theory
asked Jan 4 at 9:33
Gabriel RibeiroGabriel Ribeiro
1,456523
asked Jan 4 at 9:33
Gabriel RibeiroGabriel Ribeiro
1,456523
asked Jan 4 at 9:33
Gabriel RibeiroGabriel Ribeiro
1,456523
asked Jan 4 at 9:33
Gabriel RibeiroGabriel Ribeiro
1,456523
asked Jan 4 at 9:33
Gabriel RibeiroGabriel Ribeiro
1,456523
1,456523
$begingroup$
Short answer: no.
$endgroup$
– md2perpe
Jan 4 at 11:10
$begingroup$
@md2perpe The technique I used for solving equations of the form $fT=0$ was very useful to me. Are there other similar techniques?
$endgroup$
– Gabriel Ribeiro
Jan 4 at 11:46
$begingroup$
I know no technique to solve the equation when there is some point where $f^{(k)} = 0$ for all $k = 0, 1, 2, ldots$
$endgroup$
– md2perpe
Jan 4 at 14:54
add a comment |
$begingroup$
Short answer: no.
$endgroup$
– md2perpe
Jan 4 at 11:10
$begingroup$
@md2perpe The technique I used for solving equations of the form $fT=0$ was very useful to me. Are there other similar techniques?
$endgroup$
– Gabriel Ribeiro
Jan 4 at 11:46
$begingroup$
I know no technique to solve the equation when there is some point where $f^{(k)} = 0$ for all $k = 0, 1, 2, ldots$
$endgroup$
– md2perpe
Jan 4 at 14:54
$begingroup$
Short answer: no.
$endgroup$
– md2perpe
Jan 4 at 11:10
$begingroup$
Short answer: no.
$endgroup$
– md2perpe
Jan 4 at 11:10
$begingroup$
@md2perpe The technique I used for solving equations of the form $fT=0$ was very useful to me. Are there other similar techniques?
$endgroup$
– Gabriel Ribeiro
Jan 4 at 11:46
$begingroup$
@md2perpe The technique I used for solving equations of the form $fT=0$ was very useful to me. Are there other similar techniques?
$endgroup$
– Gabriel Ribeiro
Jan 4 at 11:46
$begingroup$
I know no technique to solve the equation when there is some point where $f^{(k)} = 0$ for all $k = 0, 1, 2, ldots$
$endgroup$
– md2perpe
Jan 4 at 14:54
$begingroup$
I know no technique to solve the equation when there is some point where $f^{(k)} = 0$ for all $k = 0, 1, 2, ldots$
$endgroup$
– md2perpe
Jan 4 at 14:54
add a comment |
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$begingroup$
Short answer: no.
$endgroup$
– md2perpe
Jan 4 at 11:10
$begingroup$
@md2perpe The technique I used for solving equations of the form $fT=0$ was very useful to me. Are there other similar techniques?
$endgroup$
– Gabriel Ribeiro
Jan 4 at 11:46
$begingroup$
I know no technique to solve the equation when there is some point where $f^{(k)} = 0$ for all $k = 0, 1, 2, ldots$
$endgroup$
– md2perpe
Jan 4 at 14:54