Convergence of Poisson integral at boundary of half plane.
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My question is given $fin L^1(mathbb{R})$ and defining
$$
f_{epsilon}(u)=frac{1}{pi}int_{mathbb{R}}f(t)frac{epsilon}{(t-u)^2+epsilon^2}dt,
$$
whether $f_epsilon$ converges to $f$ in $L^1$? If not, then is it true if $f$ has compact support or for $L^p$ spaces with $p>1$?
functional-analysis harmonic-analysis
asked Nov 16 at 19:04
Mathmo
199211
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0
down vote
favorite
My question is given $fin L^1(mathbb{R})$ and defining
$$
f_{epsilon}(u)=frac{1}{pi}int_{mathbb{R}}f(t)frac{epsilon}{(t-u)^2+epsilon^2}dt,
$$
whether $f_epsilon$ converges to $f$ in $L^1$? If not, then is it true if $f$ has compact support or for $L^p$ spaces with $p>1$?
functional-analysis harmonic-analysis
asked Nov 16 at 19:04
Mathmo
199211
$f_{epsilon}rightarrow f$ holds in $L^p$ if $f in L^p$ f for some $1 le p < infty$. It can't hold for all $fin L^p$ where $p=infty$ because a general $f in L^{infty}$ cannot be approximated in $L^{infty}$ by a continuous function, which $f_{epsilon}$ is for $epsilon > 0$.
– DisintegratingByParts
Nov 16 at 19:11
Great, thanks! Do you have a reference for this?
– Mathmo
Nov 16 at 20:45
I seem to recall seeing this covered in Duren's book on $H^p$ spaces.
– DisintegratingByParts
Nov 16 at 21:02
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
My question is given $fin L^1(mathbb{R})$ and defining
$$
f_{epsilon}(u)=frac{1}{pi}int_{mathbb{R}}f(t)frac{epsilon}{(t-u)^2+epsilon^2}dt,
$$
whether $f_epsilon$ converges to $f$ in $L^1$? If not, then is it true if $f$ has compact support or for $L^p$ spaces with $p>1$?
functional-analysis harmonic-analysis
asked Nov 16 at 19:04
Mathmo
199211
My question is given $fin L^1(mathbb{R})$ and defining
$$
f_{epsilon}(u)=frac{1}{pi}int_{mathbb{R}}f(t)frac{epsilon}{(t-u)^2+epsilon^2}dt,
$$
whether $f_epsilon$ converges to $f$ in $L^1$? If not, then is it true if $f$ has compact support or for $L^p$ spaces with $p>1$?
functional-analysis harmonic-analysis
functional-analysis harmonic-analysis
asked Nov 16 at 19:04
Mathmo
199211
asked Nov 16 at 19:04
Mathmo
199211
asked Nov 16 at 19:04
Mathmo
199211
asked Nov 16 at 19:04
Mathmo
199211
asked Nov 16 at 19:04
Mathmo
199211
199211
$f_{epsilon}rightarrow f$ holds in $L^p$ if $f in L^p$ f for some $1 le p < infty$. It can't hold for all $fin L^p$ where $p=infty$ because a general $f in L^{infty}$ cannot be approximated in $L^{infty}$ by a continuous function, which $f_{epsilon}$ is for $epsilon > 0$.
– DisintegratingByParts
Nov 16 at 19:11
Great, thanks! Do you have a reference for this?
– Mathmo
Nov 16 at 20:45
I seem to recall seeing this covered in Duren's book on $H^p$ spaces.
– DisintegratingByParts
Nov 16 at 21:02
add a comment |
$f_{epsilon}rightarrow f$ holds in $L^p$ if $f in L^p$ f for some $1 le p < infty$. It can't hold for all $fin L^p$ where $p=infty$ because a general $f in L^{infty}$ cannot be approximated in $L^{infty}$ by a continuous function, which $f_{epsilon}$ is for $epsilon > 0$.
– DisintegratingByParts
Nov 16 at 19:11
Great, thanks! Do you have a reference for this?
– Mathmo
Nov 16 at 20:45
I seem to recall seeing this covered in Duren's book on $H^p$ spaces.
– DisintegratingByParts
Nov 16 at 21:02
$f_{epsilon}rightarrow f$ holds in $L^p$ if $f in L^p$ f for some $1 le p < infty$. It can't hold for all $fin L^p$ where $p=infty$ because a general $f in L^{infty}$ cannot be approximated in $L^{infty}$ by a continuous function, which $f_{epsilon}$ is for $epsilon > 0$.
– DisintegratingByParts
Nov 16 at 19:11
$f_{epsilon}rightarrow f$ holds in $L^p$ if $f in L^p$ f for some $1 le p < infty$. It can't hold for all $fin L^p$ where $p=infty$ because a general $f in L^{infty}$ cannot be approximated in $L^{infty}$ by a continuous function, which $f_{epsilon}$ is for $epsilon > 0$.
– DisintegratingByParts
Nov 16 at 19:11
Great, thanks! Do you have a reference for this?
– Mathmo
Nov 16 at 20:45
Great, thanks! Do you have a reference for this?
– Mathmo
Nov 16 at 20:45
I seem to recall seeing this covered in Duren's book on $H^p$ spaces.
– DisintegratingByParts
Nov 16 at 21:02
I seem to recall seeing this covered in Duren's book on $H^p$ spaces.
– DisintegratingByParts
Nov 16 at 21:02
add a comment |
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$f_{epsilon}rightarrow f$ holds in $L^p$ if $f in L^p$ f for some $1 le p < infty$. It can't hold for all $fin L^p$ where $p=infty$ because a general $f in L^{infty}$ cannot be approximated in $L^{infty}$ by a continuous function, which $f_{epsilon}$ is for $epsilon > 0$.
– DisintegratingByParts
Nov 16 at 19:11
Great, thanks! Do you have a reference for this?
– Mathmo
Nov 16 at 20:45
I seem to recall seeing this covered in Duren's book on $H^p$ spaces.
– DisintegratingByParts
Nov 16 at 21:02