Decay of positive definite functions in Lp
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Let $f:mathbb{R}rightarrowmathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means
$$
sum_{i=1}^{n}sum_{j=1}^{n}f(x_i-x_j)y_i y_j geq 0
$$
for all $ngeq 1, x,yin mathbb{R}^n$.
Question. If $f in L^{p}$ for some $p>2$, must we have $f(x)=O(|x|^{-c})$ for some $c>0$?
Note that, by Bochner's theorem, $f = widehat{mu}$ for some probability measure $mu$ on $mathbb{R}$.
Context: This is a natural extension of this question: $L^p$ implies polynomial decay?
I am posting this over at MO, since it has stood several months here with no progress. https://mathoverflow.net/questions/278988/decay-of-positive-definite-function-in-lp
real-analysis fourier-analysis harmonic-analysis
asked Jun 1 '17 at 19:31
RitterSportRitterSport
1,017315
$endgroup$
add a comment |
$begingroup$
Let $f:mathbb{R}rightarrowmathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means
$$
sum_{i=1}^{n}sum_{j=1}^{n}f(x_i-x_j)y_i y_j geq 0
$$
for all $ngeq 1, x,yin mathbb{R}^n$.
Question. If $f in L^{p}$ for some $p>2$, must we have $f(x)=O(|x|^{-c})$ for some $c>0$?
Note that, by Bochner's theorem, $f = widehat{mu}$ for some probability measure $mu$ on $mathbb{R}$.
Context: This is a natural extension of this question: $L^p$ implies polynomial decay?
I am posting this over at MO, since it has stood several months here with no progress. https://mathoverflow.net/questions/278988/decay-of-positive-definite-function-in-lp
real-analysis fourier-analysis harmonic-analysis
asked Jun 1 '17 at 19:31
RitterSportRitterSport
1,017315
$endgroup$
$begingroup$
It's been a while since I read up on them, but if I recall correctly, the maximum value is attained at 0, right?
$endgroup$
– Cameron Williams
Jul 20 '17 at 16:24
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@CameronWilliams That's true.
$endgroup$
– RitterSport
Jul 20 '17 at 16:25
$begingroup$
The question was answered at Mathoverflow.
$endgroup$
– Davide Giraudo
Dec 11 '18 at 20:40
add a comment |
$begingroup$
Let $f:mathbb{R}rightarrowmathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means
$$
sum_{i=1}^{n}sum_{j=1}^{n}f(x_i-x_j)y_i y_j geq 0
$$
for all $ngeq 1, x,yin mathbb{R}^n$.
Question. If $f in L^{p}$ for some $p>2$, must we have $f(x)=O(|x|^{-c})$ for some $c>0$?
Note that, by Bochner's theorem, $f = widehat{mu}$ for some probability measure $mu$ on $mathbb{R}$.
Context: This is a natural extension of this question: $L^p$ implies polynomial decay?
I am posting this over at MO, since it has stood several months here with no progress. https://mathoverflow.net/questions/278988/decay-of-positive-definite-function-in-lp
real-analysis fourier-analysis harmonic-analysis
asked Jun 1 '17 at 19:31
RitterSportRitterSport
1,017315
$endgroup$
Let $f:mathbb{R}rightarrowmathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means
$$
sum_{i=1}^{n}sum_{j=1}^{n}f(x_i-x_j)y_i y_j geq 0
$$
for all $ngeq 1, x,yin mathbb{R}^n$.
Question. If $f in L^{p}$ for some $p>2$, must we have $f(x)=O(|x|^{-c})$ for some $c>0$?
Note that, by Bochner's theorem, $f = widehat{mu}$ for some probability measure $mu$ on $mathbb{R}$.
Context: This is a natural extension of this question: $L^p$ implies polynomial decay?
I am posting this over at MO, since it has stood several months here with no progress. https://mathoverflow.net/questions/278988/decay-of-positive-definite-function-in-lp
real-analysis fourier-analysis harmonic-analysis
real-analysis fourier-analysis harmonic-analysis
asked Jun 1 '17 at 19:31
RitterSportRitterSport
1,017315
asked Jun 1 '17 at 19:31
RitterSportRitterSport
1,017315
edited Aug 17 '17 at 21:26
RitterSport
asked Jun 1 '17 at 19:31
RitterSportRitterSport
1,017315
asked Jun 1 '17 at 19:31
RitterSportRitterSport
1,017315
asked Jun 1 '17 at 19:31
RitterSportRitterSport
1,017315
1,017315
$begingroup$
It's been a while since I read up on them, but if I recall correctly, the maximum value is attained at 0, right?
$endgroup$
– Cameron Williams
Jul 20 '17 at 16:24
$begingroup$
@CameronWilliams That's true.
$endgroup$
– RitterSport
Jul 20 '17 at 16:25
$begingroup$
The question was answered at Mathoverflow.
$endgroup$
– Davide Giraudo
Dec 11 '18 at 20:40
add a comment |
$begingroup$
It's been a while since I read up on them, but if I recall correctly, the maximum value is attained at 0, right?
$endgroup$
– Cameron Williams
Jul 20 '17 at 16:24
$begingroup$
@CameronWilliams That's true.
$endgroup$
– RitterSport
Jul 20 '17 at 16:25
$begingroup$
The question was answered at Mathoverflow.
$endgroup$
– Davide Giraudo
Dec 11 '18 at 20:40
$begingroup$
It's been a while since I read up on them, but if I recall correctly, the maximum value is attained at 0, right?
$endgroup$
– Cameron Williams
Jul 20 '17 at 16:24
$begingroup$
It's been a while since I read up on them, but if I recall correctly, the maximum value is attained at 0, right?
$endgroup$
– Cameron Williams
Jul 20 '17 at 16:24
$begingroup$
@CameronWilliams That's true.
$endgroup$
– RitterSport
Jul 20 '17 at 16:25
$begingroup$
@CameronWilliams That's true.
$endgroup$
– RitterSport
Jul 20 '17 at 16:25
$begingroup$
The question was answered at Mathoverflow.
$endgroup$
– Davide Giraudo
Dec 11 '18 at 20:40
$begingroup$
The question was answered at Mathoverflow.
$endgroup$
– Davide Giraudo
Dec 11 '18 at 20:40
add a comment |
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$begingroup$
It's been a while since I read up on them, but if I recall correctly, the maximum value is attained at 0, right?
$endgroup$
– Cameron Williams
Jul 20 '17 at 16:24
$begingroup$
@CameronWilliams That's true.
$endgroup$
– RitterSport
Jul 20 '17 at 16:25
$begingroup$
The question was answered at Mathoverflow.
$endgroup$
– Davide Giraudo
Dec 11 '18 at 20:40