Mixing
Obtain a function from local mean values [closed]
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(1) Suppose I have a (bounded, measurable) function $f$ on a compact subset $K$ of $R^n$, and for every open interval $Usubset K$ I know only the mean value
$$
m_U := frac{1}{|U|} int_U f dx.
$$
What additional constraints are required to uniquely recover $f$ almost everywhere? Or: what is the largest space of functions you can think of which allows this?
(2) Same situation, but I know the mean not for every $U$, but only nested sequences of intervals, in the sense of a finer and finer discretization of $K$. For simplicity, let's assume $K=[0,1]$ and $U$ is of the form $( k/2^n, (k+1) / 2^n)$ with $n geq 0$ and $0 leq k leq 2^n-1$.
If you can point me to theorems/sections of books which might yield an answer to the questions or sketch an idea for a possible answer and proof, I would be most happy.
functional-analysis measure-theory
asked Jan 12 at 18:34
NyarlathotepNyarlathotep
111
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closed as off-topic by Adrian Keister, Cesareo, Paul Frost, Pierre-Guy Plamondon, Aweygan Jan 13 at 16:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Adrian Keister, Cesareo, Paul Frost, Pierre-Guy Plamondon, Aweygan
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
(1) Suppose I have a (bounded, measurable) function $f$ on a compact subset $K$ of $R^n$, and for every open interval $Usubset K$ I know only the mean value
$$
m_U := frac{1}{|U|} int_U f dx.
$$
What additional constraints are required to uniquely recover $f$ almost everywhere? Or: what is the largest space of functions you can think of which allows this?
(2) Same situation, but I know the mean not for every $U$, but only nested sequences of intervals, in the sense of a finer and finer discretization of $K$. For simplicity, let's assume $K=[0,1]$ and $U$ is of the form $( k/2^n, (k+1) / 2^n)$ with $n geq 0$ and $0 leq k leq 2^n-1$.
If you can point me to theorems/sections of books which might yield an answer to the questions or sketch an idea for a possible answer and proof, I would be most happy.
functional-analysis measure-theory
asked Jan 12 at 18:34
NyarlathotepNyarlathotep
111
$endgroup$
closed as off-topic by Adrian Keister, Cesareo, Paul Frost, Pierre-Guy Plamondon, Aweygan Jan 13 at 16:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Adrian Keister, Cesareo, Paul Frost, Pierre-Guy Plamondon, Aweygan
If this question can be reworded to fit the rules in the help center, please edit the question.
2
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This theorem deals exactly with your question: en.wikipedia.org/wiki/Lebesgue_differentiation_theorem. Hope it helps
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– pitariver
Jan 12 at 18:39
$begingroup$
Exactly what I was looking for, thanks much !
$endgroup$
– Nyarlathotep
Jan 12 at 18:56
add a comment |
$begingroup$
(1) Suppose I have a (bounded, measurable) function $f$ on a compact subset $K$ of $R^n$, and for every open interval $Usubset K$ I know only the mean value
$$
m_U := frac{1}{|U|} int_U f dx.
$$
What additional constraints are required to uniquely recover $f$ almost everywhere? Or: what is the largest space of functions you can think of which allows this?
(2) Same situation, but I know the mean not for every $U$, but only nested sequences of intervals, in the sense of a finer and finer discretization of $K$. For simplicity, let's assume $K=[0,1]$ and $U$ is of the form $( k/2^n, (k+1) / 2^n)$ with $n geq 0$ and $0 leq k leq 2^n-1$.
If you can point me to theorems/sections of books which might yield an answer to the questions or sketch an idea for a possible answer and proof, I would be most happy.
functional-analysis measure-theory
asked Jan 12 at 18:34
NyarlathotepNyarlathotep
111
$endgroup$
(1) Suppose I have a (bounded, measurable) function $f$ on a compact subset $K$ of $R^n$, and for every open interval $Usubset K$ I know only the mean value
$$
m_U := frac{1}{|U|} int_U f dx.
$$
What additional constraints are required to uniquely recover $f$ almost everywhere? Or: what is the largest space of functions you can think of which allows this?
(2) Same situation, but I know the mean not for every $U$, but only nested sequences of intervals, in the sense of a finer and finer discretization of $K$. For simplicity, let's assume $K=[0,1]$ and $U$ is of the form $( k/2^n, (k+1) / 2^n)$ with $n geq 0$ and $0 leq k leq 2^n-1$.
If you can point me to theorems/sections of books which might yield an answer to the questions or sketch an idea for a possible answer and proof, I would be most happy.
functional-analysis measure-theory
functional-analysis measure-theory
asked Jan 12 at 18:34
NyarlathotepNyarlathotep
111
asked Jan 12 at 18:34
NyarlathotepNyarlathotep
111
asked Jan 12 at 18:34
NyarlathotepNyarlathotep
111
asked Jan 12 at 18:34
NyarlathotepNyarlathotep
111
asked Jan 12 at 18:34
NyarlathotepNyarlathotep
111
111
closed as off-topic by Adrian Keister, Cesareo, Paul Frost, Pierre-Guy Plamondon, Aweygan Jan 13 at 16:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Adrian Keister, Cesareo, Paul Frost, Pierre-Guy Plamondon, Aweygan
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Adrian Keister, Cesareo, Paul Frost, Pierre-Guy Plamondon, Aweygan Jan 13 at 16:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Adrian Keister, Cesareo, Paul Frost, Pierre-Guy Plamondon, Aweygan
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
This theorem deals exactly with your question: en.wikipedia.org/wiki/Lebesgue_differentiation_theorem. Hope it helps
$endgroup$
– pitariver
Jan 12 at 18:39
$begingroup$
Exactly what I was looking for, thanks much !
$endgroup$
– Nyarlathotep
Jan 12 at 18:56
add a comment |
2
$begingroup$
This theorem deals exactly with your question: en.wikipedia.org/wiki/Lebesgue_differentiation_theorem. Hope it helps
$endgroup$
– pitariver
Jan 12 at 18:39
$begingroup$
Exactly what I was looking for, thanks much !
$endgroup$
– Nyarlathotep
Jan 12 at 18:56
2
2
$begingroup$
This theorem deals exactly with your question: en.wikipedia.org/wiki/Lebesgue_differentiation_theorem. Hope it helps
$endgroup$
– pitariver
Jan 12 at 18:39
$begingroup$
This theorem deals exactly with your question: en.wikipedia.org/wiki/Lebesgue_differentiation_theorem. Hope it helps
$endgroup$
– pitariver
Jan 12 at 18:39
$begingroup$
Exactly what I was looking for, thanks much !
$endgroup$
– Nyarlathotep
Jan 12 at 18:56
$begingroup$
Exactly what I was looking for, thanks much !
$endgroup$
– Nyarlathotep
Jan 12 at 18:56
add a comment |
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2
$begingroup$
This theorem deals exactly with your question: en.wikipedia.org/wiki/Lebesgue_differentiation_theorem. Hope it helps
$endgroup$
– pitariver
Jan 12 at 18:39
$begingroup$
Exactly what I was looking for, thanks much !
$endgroup$
– Nyarlathotep
Jan 12 at 18:56