Clarification required for the proof of Riemann Lebesgue Lemma
1
$begingroup$
Riemann Lebesgue Lemma Proof.
In the proof given above, I came across an unfamiliar notation $g= sum m_i chi_{[x_i-1,x_i]}$. What does the $chi$ mean here? Is it the characteristic function of the set $[x_i-1,x_i]$?
Please clarify in detail.
proof-explanation riemann-integration
asked Jan 17 at 10:23
Subhasis BiswasSubhasis Biswas
493311
$endgroup$
|
show 3 more comments
1
$begingroup$
Riemann Lebesgue Lemma Proof.
In the proof given above, I came across an unfamiliar notation $g= sum m_i chi_{[x_i-1,x_i]}$. What does the $chi$ mean here? Is it the characteristic function of the set $[x_i-1,x_i]$?
Please clarify in detail.
proof-explanation riemann-integration
asked Jan 17 at 10:23
Subhasis BiswasSubhasis Biswas
493311
$endgroup$
2
$begingroup$
Yes, it is the characteristic function.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:24
$begingroup$
what role does it play in here?
$endgroup$
– Subhasis Biswas
Jan 17 at 10:25
1
$begingroup$
It means $g$ is a step function: when $x in [x_{j-1}, x_j]$, $g(x) = m_j$.
$endgroup$
– xbh
Jan 17 at 10:29
2
$begingroup$
R-L Lemma is trivial when when the function you start with is the characteristic function of some closed interval. [Verify this!]. Hence it is true for linear combinations of such functions. The basic idea of the proof of R-L Lemma is to approximate the given function by a linear combination of characteristic functions of closed intervals.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:31
1
$begingroup$
Yes. The appearance of $lambda $ in the denominator is what makes the whole things work.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 11:46
|
show 3 more comments
1
1
1
$begingroup$
Riemann Lebesgue Lemma Proof.
In the proof given above, I came across an unfamiliar notation $g= sum m_i chi_{[x_i-1,x_i]}$. What does the $chi$ mean here? Is it the characteristic function of the set $[x_i-1,x_i]$?
Please clarify in detail.
proof-explanation riemann-integration
asked Jan 17 at 10:23
Subhasis BiswasSubhasis Biswas
493311
$endgroup$
Riemann Lebesgue Lemma Proof.
In the proof given above, I came across an unfamiliar notation $g= sum m_i chi_{[x_i-1,x_i]}$. What does the $chi$ mean here? Is it the characteristic function of the set $[x_i-1,x_i]$?
Please clarify in detail.
proof-explanation riemann-integration
proof-explanation riemann-integration
asked Jan 17 at 10:23
Subhasis BiswasSubhasis Biswas
493311
asked Jan 17 at 10:23
Subhasis BiswasSubhasis Biswas
493311
asked Jan 17 at 10:23
Subhasis BiswasSubhasis Biswas
493311
asked Jan 17 at 10:23
Subhasis BiswasSubhasis Biswas
493311
asked Jan 17 at 10:23
Subhasis BiswasSubhasis Biswas
493311
493311
2
$begingroup$
Yes, it is the characteristic function.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:24
$begingroup$
what role does it play in here?
$endgroup$
– Subhasis Biswas
Jan 17 at 10:25
1
$begingroup$
It means $g$ is a step function: when $x in [x_{j-1}, x_j]$, $g(x) = m_j$.
$endgroup$
– xbh
Jan 17 at 10:29
2
$begingroup$
R-L Lemma is trivial when when the function you start with is the characteristic function of some closed interval. [Verify this!]. Hence it is true for linear combinations of such functions. The basic idea of the proof of R-L Lemma is to approximate the given function by a linear combination of characteristic functions of closed intervals.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:31
1
$begingroup$
Yes. The appearance of $lambda $ in the denominator is what makes the whole things work.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 11:46
|
show 3 more comments
2
$begingroup$
Yes, it is the characteristic function.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:24
$begingroup$
what role does it play in here?
$endgroup$
– Subhasis Biswas
Jan 17 at 10:25
1
$begingroup$
It means $g$ is a step function: when $x in [x_{j-1}, x_j]$, $g(x) = m_j$.
$endgroup$
– xbh
Jan 17 at 10:29
2
$begingroup$
R-L Lemma is trivial when when the function you start with is the characteristic function of some closed interval. [Verify this!]. Hence it is true for linear combinations of such functions. The basic idea of the proof of R-L Lemma is to approximate the given function by a linear combination of characteristic functions of closed intervals.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:31
1
$begingroup$
Yes. The appearance of $lambda $ in the denominator is what makes the whole things work.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 11:46
2
2
$begingroup$
Yes, it is the characteristic function.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:24
$begingroup$
Yes, it is the characteristic function.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:24
$begingroup$
what role does it play in here?
$endgroup$
– Subhasis Biswas
Jan 17 at 10:25
$begingroup$
what role does it play in here?
$endgroup$
– Subhasis Biswas
Jan 17 at 10:25
1
1
$begingroup$
It means $g$ is a step function: when $x in [x_{j-1}, x_j]$, $g(x) = m_j$.
$endgroup$
– xbh
Jan 17 at 10:29
$begingroup$
It means $g$ is a step function: when $x in [x_{j-1}, x_j]$, $g(x) = m_j$.
$endgroup$
– xbh
Jan 17 at 10:29
2
2
$begingroup$
R-L Lemma is trivial when when the function you start with is the characteristic function of some closed interval. [Verify this!]. Hence it is true for linear combinations of such functions. The basic idea of the proof of R-L Lemma is to approximate the given function by a linear combination of characteristic functions of closed intervals.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:31
$begingroup$
R-L Lemma is trivial when when the function you start with is the characteristic function of some closed interval. [Verify this!]. Hence it is true for linear combinations of such functions. The basic idea of the proof of R-L Lemma is to approximate the given function by a linear combination of characteristic functions of closed intervals.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:31
1
1
$begingroup$
Yes. The appearance of $lambda $ in the denominator is what makes the whole things work.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 11:46
$begingroup$
Yes. The appearance of $lambda $ in the denominator is what makes the whole things work.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 11:46
|
show 3 more comments
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2
$begingroup$
Yes, it is the characteristic function.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:24
$begingroup$
what role does it play in here?
$endgroup$
– Subhasis Biswas
Jan 17 at 10:25
1
$begingroup$
It means $g$ is a step function: when $x in [x_{j-1}, x_j]$, $g(x) = m_j$.
$endgroup$
– xbh
Jan 17 at 10:29
2
$begingroup$
R-L Lemma is trivial when when the function you start with is the characteristic function of some closed interval. [Verify this!]. Hence it is true for linear combinations of such functions. The basic idea of the proof of R-L Lemma is to approximate the given function by a linear combination of characteristic functions of closed intervals.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 10:31
1
$begingroup$
Yes. The appearance of $lambda $ in the denominator is what makes the whole things work.
$endgroup$
– Kavi Rama Murthy
Jan 17 at 11:46