Support of a fundamental solution of wave equation
$begingroup$
I need to prove that the following fundamental solution of the wave equation:
$$ E(x,t)= mathfrak{F}^{-1}Bigg(frac{sin ( t|.|)}{|.|}Bigg)frac{theta(t)}{(2pi)^{n/2}}(x) in mathcal{S}'(mathbb{R}^{1,n})$$
satisfies:
$$ supp(E) subset { (t,x) in mathbb{R}^{1,n}: |x| leq t}$$
(where $mathfrak{F}^{-1}$ - inverse Fourie transform wrt. "spatial" coordinates).
It is also hinted that Paley-Wiener-Schwartz theorem is used in this.
I have found a theorem that states the following:
Let $u in mathcal{D}'(mathbb{R}^n)$ and $f in C^infty(mathbb{R}^n)$. If $fu=0$ then $$ supp(u) subset { xin mathbb{R}^n | f(x)=0} $$
Is this theorem somehow applicable here?
pde fourier-analysis harmonic-analysis
asked Jan 14 at 17:24
Sergey DyldaSergey Dylda
1616
$endgroup$
add a comment |
$begingroup$
I need to prove that the following fundamental solution of the wave equation:
$$ E(x,t)= mathfrak{F}^{-1}Bigg(frac{sin ( t|.|)}{|.|}Bigg)frac{theta(t)}{(2pi)^{n/2}}(x) in mathcal{S}'(mathbb{R}^{1,n})$$
satisfies:
$$ supp(E) subset { (t,x) in mathbb{R}^{1,n}: |x| leq t}$$
(where $mathfrak{F}^{-1}$ - inverse Fourie transform wrt. "spatial" coordinates).
It is also hinted that Paley-Wiener-Schwartz theorem is used in this.
I have found a theorem that states the following:
Let $u in mathcal{D}'(mathbb{R}^n)$ and $f in C^infty(mathbb{R}^n)$. If $fu=0$ then $$ supp(u) subset { xin mathbb{R}^n | f(x)=0} $$
Is this theorem somehow applicable here?
pde fourier-analysis harmonic-analysis
asked Jan 14 at 17:24
Sergey DyldaSergey Dylda
1616
$endgroup$
add a comment |
$begingroup$
I need to prove that the following fundamental solution of the wave equation:
$$ E(x,t)= mathfrak{F}^{-1}Bigg(frac{sin ( t|.|)}{|.|}Bigg)frac{theta(t)}{(2pi)^{n/2}}(x) in mathcal{S}'(mathbb{R}^{1,n})$$
satisfies:
$$ supp(E) subset { (t,x) in mathbb{R}^{1,n}: |x| leq t}$$
(where $mathfrak{F}^{-1}$ - inverse Fourie transform wrt. "spatial" coordinates).
It is also hinted that Paley-Wiener-Schwartz theorem is used in this.
I have found a theorem that states the following:
Let $u in mathcal{D}'(mathbb{R}^n)$ and $f in C^infty(mathbb{R}^n)$. If $fu=0$ then $$ supp(u) subset { xin mathbb{R}^n | f(x)=0} $$
Is this theorem somehow applicable here?
pde fourier-analysis harmonic-analysis
asked Jan 14 at 17:24
Sergey DyldaSergey Dylda
1616
$endgroup$
I need to prove that the following fundamental solution of the wave equation:
$$ E(x,t)= mathfrak{F}^{-1}Bigg(frac{sin ( t|.|)}{|.|}Bigg)frac{theta(t)}{(2pi)^{n/2}}(x) in mathcal{S}'(mathbb{R}^{1,n})$$
satisfies:
$$ supp(E) subset { (t,x) in mathbb{R}^{1,n}: |x| leq t}$$
(where $mathfrak{F}^{-1}$ - inverse Fourie transform wrt. "spatial" coordinates).
It is also hinted that Paley-Wiener-Schwartz theorem is used in this.
I have found a theorem that states the following:
Let $u in mathcal{D}'(mathbb{R}^n)$ and $f in C^infty(mathbb{R}^n)$. If $fu=0$ then $$ supp(u) subset { xin mathbb{R}^n | f(x)=0} $$
Is this theorem somehow applicable here?
pde fourier-analysis harmonic-analysis
pde fourier-analysis harmonic-analysis
asked Jan 14 at 17:24
Sergey DyldaSergey Dylda
1616
asked Jan 14 at 17:24
Sergey DyldaSergey Dylda
1616
edited Jan 14 at 17:35
Sergey Dylda
asked Jan 14 at 17:24
Sergey DyldaSergey Dylda
1616
asked Jan 14 at 17:24
Sergey DyldaSergey Dylda
1616
asked Jan 14 at 17:24
Sergey DyldaSergey Dylda
1616
1616
add a comment |
add a comment |
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