What is the definition of function of bounded variation over the whole interval $(-infty, +infty)$?
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What is the definition of real-valued function of bounded variation over the whole interval $(-infty, +infty)$? I know one definition of functions of bounded variation, but the interval here is $[a,b]subset mathbb R$. How to set a partition of $(-infty, +infty)$? ${-infty=x_0, x_1, dots, x_n=+infty}$?
real-analysis analysis
asked Dec 2 '18 at 5:07
user398843user398843
636215
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add a comment |
$begingroup$
What is the definition of real-valued function of bounded variation over the whole interval $(-infty, +infty)$? I know one definition of functions of bounded variation, but the interval here is $[a,b]subset mathbb R$. How to set a partition of $(-infty, +infty)$? ${-infty=x_0, x_1, dots, x_n=+infty}$?
real-analysis analysis
asked Dec 2 '18 at 5:07
user398843user398843
636215
$endgroup$
add a comment |
$begingroup$
What is the definition of real-valued function of bounded variation over the whole interval $(-infty, +infty)$? I know one definition of functions of bounded variation, but the interval here is $[a,b]subset mathbb R$. How to set a partition of $(-infty, +infty)$? ${-infty=x_0, x_1, dots, x_n=+infty}$?
real-analysis analysis
asked Dec 2 '18 at 5:07
user398843user398843
636215
$endgroup$
What is the definition of real-valued function of bounded variation over the whole interval $(-infty, +infty)$? I know one definition of functions of bounded variation, but the interval here is $[a,b]subset mathbb R$. How to set a partition of $(-infty, +infty)$? ${-infty=x_0, x_1, dots, x_n=+infty}$?
real-analysis analysis
real-analysis analysis
asked Dec 2 '18 at 5:07
user398843user398843
636215
asked Dec 2 '18 at 5:07
user398843user398843
636215
asked Dec 2 '18 at 5:07
user398843user398843
636215
asked Dec 2 '18 at 5:07
user398843user398843
636215
asked Dec 2 '18 at 5:07
user398843user398843
636215
636215
add a comment |
add a comment |
1 Answer
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Let $F:Bbb Rrightarrow Bbb R$ be a function. For each $xin Bbb R$ define $$T_F(x):=sup{sum_1^n|F(x_j)-F(x_{j-1})|:nin Bbb N ,-infty<x_0<.....<x_n=x}.$$ The function $T_F$ is increasing with values in $[0,infty]$. So the limit $T_F(infty):=lim_{xrightarrow infty} T_F(x)$ exists. If this limit is finite then we say $F$ is of bounded variation on $Bbb R$.
answered Dec 2 '18 at 5:16
UserSUserS
1,5391112
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This isn't a natural definition, since, with this, even something like $f(x) = x$ would fail to have bounded variation. We want the notion of bounded variation to capture "well-behavedness" in some sense, and certainly the identity function is well-behaved.
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– MathematicsStudent1122
Dec 2 '18 at 5:21
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Note one thing, book like Real analysis modern techniques written by Folland follows this definition. Actually this definition is an extended version of the definition of bounded variation in compact interval. Another fact is that, for a Lebesgue integrable function $f$ on real line the function $F(x)=int_{-infty}^x f(t)dt$ satisfies the condition given in definition, as we do for a Lebesgue integrable function in compact interval.
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– UserS
Dec 2 '18 at 5:59
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Let $F:Bbb Rrightarrow Bbb R$ be a function. For each $xin Bbb R$ define $$T_F(x):=sup{sum_1^n|F(x_j)-F(x_{j-1})|:nin Bbb N ,-infty<x_0<.....<x_n=x}.$$ The function $T_F$ is increasing with values in $[0,infty]$. So the limit $T_F(infty):=lim_{xrightarrow infty} T_F(x)$ exists. If this limit is finite then we say $F$ is of bounded variation on $Bbb R$.
answered Dec 2 '18 at 5:16
UserSUserS
1,5391112
$endgroup$
$begingroup$
This isn't a natural definition, since, with this, even something like $f(x) = x$ would fail to have bounded variation. We want the notion of bounded variation to capture "well-behavedness" in some sense, and certainly the identity function is well-behaved.
$endgroup$
– MathematicsStudent1122
Dec 2 '18 at 5:21
$begingroup$
Note one thing, book like Real analysis modern techniques written by Folland follows this definition. Actually this definition is an extended version of the definition of bounded variation in compact interval. Another fact is that, for a Lebesgue integrable function $f$ on real line the function $F(x)=int_{-infty}^x f(t)dt$ satisfies the condition given in definition, as we do for a Lebesgue integrable function in compact interval.
$endgroup$
– UserS
Dec 2 '18 at 5:59
add a comment |
$begingroup$
Let $F:Bbb Rrightarrow Bbb R$ be a function. For each $xin Bbb R$ define $$T_F(x):=sup{sum_1^n|F(x_j)-F(x_{j-1})|:nin Bbb N ,-infty<x_0<.....<x_n=x}.$$ The function $T_F$ is increasing with values in $[0,infty]$. So the limit $T_F(infty):=lim_{xrightarrow infty} T_F(x)$ exists. If this limit is finite then we say $F$ is of bounded variation on $Bbb R$.
answered Dec 2 '18 at 5:16
UserSUserS
1,5391112
$endgroup$
$begingroup$
This isn't a natural definition, since, with this, even something like $f(x) = x$ would fail to have bounded variation. We want the notion of bounded variation to capture "well-behavedness" in some sense, and certainly the identity function is well-behaved.
$endgroup$
– MathematicsStudent1122
Dec 2 '18 at 5:21
$begingroup$
Note one thing, book like Real analysis modern techniques written by Folland follows this definition. Actually this definition is an extended version of the definition of bounded variation in compact interval. Another fact is that, for a Lebesgue integrable function $f$ on real line the function $F(x)=int_{-infty}^x f(t)dt$ satisfies the condition given in definition, as we do for a Lebesgue integrable function in compact interval.
$endgroup$
– UserS
Dec 2 '18 at 5:59
add a comment |
$begingroup$
Let $F:Bbb Rrightarrow Bbb R$ be a function. For each $xin Bbb R$ define $$T_F(x):=sup{sum_1^n|F(x_j)-F(x_{j-1})|:nin Bbb N ,-infty<x_0<.....<x_n=x}.$$ The function $T_F$ is increasing with values in $[0,infty]$. So the limit $T_F(infty):=lim_{xrightarrow infty} T_F(x)$ exists. If this limit is finite then we say $F$ is of bounded variation on $Bbb R$.
answered Dec 2 '18 at 5:16
UserSUserS
1,5391112
$endgroup$
Let $F:Bbb Rrightarrow Bbb R$ be a function. For each $xin Bbb R$ define $$T_F(x):=sup{sum_1^n|F(x_j)-F(x_{j-1})|:nin Bbb N ,-infty<x_0<.....<x_n=x}.$$ The function $T_F$ is increasing with values in $[0,infty]$. So the limit $T_F(infty):=lim_{xrightarrow infty} T_F(x)$ exists. If this limit is finite then we say $F$ is of bounded variation on $Bbb R$.
answered Dec 2 '18 at 5:16
UserSUserS
1,5391112
answered Dec 2 '18 at 5:16
UserSUserS
1,5391112
answered Dec 2 '18 at 5:16
UserSUserS
1,5391112
answered Dec 2 '18 at 5:16
UserSUserS
1,5391112
1,5391112
$begingroup$
This isn't a natural definition, since, with this, even something like $f(x) = x$ would fail to have bounded variation. We want the notion of bounded variation to capture "well-behavedness" in some sense, and certainly the identity function is well-behaved.
$endgroup$
– MathematicsStudent1122
Dec 2 '18 at 5:21
$begingroup$
Note one thing, book like Real analysis modern techniques written by Folland follows this definition. Actually this definition is an extended version of the definition of bounded variation in compact interval. Another fact is that, for a Lebesgue integrable function $f$ on real line the function $F(x)=int_{-infty}^x f(t)dt$ satisfies the condition given in definition, as we do for a Lebesgue integrable function in compact interval.
$endgroup$
– UserS
Dec 2 '18 at 5:59
add a comment |
$begingroup$
This isn't a natural definition, since, with this, even something like $f(x) = x$ would fail to have bounded variation. We want the notion of bounded variation to capture "well-behavedness" in some sense, and certainly the identity function is well-behaved.
$endgroup$
– MathematicsStudent1122
Dec 2 '18 at 5:21
$begingroup$
Note one thing, book like Real analysis modern techniques written by Folland follows this definition. Actually this definition is an extended version of the definition of bounded variation in compact interval. Another fact is that, for a Lebesgue integrable function $f$ on real line the function $F(x)=int_{-infty}^x f(t)dt$ satisfies the condition given in definition, as we do for a Lebesgue integrable function in compact interval.
$endgroup$
– UserS
Dec 2 '18 at 5:59
$begingroup$
This isn't a natural definition, since, with this, even something like $f(x) = x$ would fail to have bounded variation. We want the notion of bounded variation to capture "well-behavedness" in some sense, and certainly the identity function is well-behaved.
$endgroup$
– MathematicsStudent1122
Dec 2 '18 at 5:21
$begingroup$
This isn't a natural definition, since, with this, even something like $f(x) = x$ would fail to have bounded variation. We want the notion of bounded variation to capture "well-behavedness" in some sense, and certainly the identity function is well-behaved.
$endgroup$
– MathematicsStudent1122
Dec 2 '18 at 5:21
$begingroup$
Note one thing, book like Real analysis modern techniques written by Folland follows this definition. Actually this definition is an extended version of the definition of bounded variation in compact interval. Another fact is that, for a Lebesgue integrable function $f$ on real line the function $F(x)=int_{-infty}^x f(t)dt$ satisfies the condition given in definition, as we do for a Lebesgue integrable function in compact interval.
$endgroup$
– UserS
Dec 2 '18 at 5:59
$begingroup$
Note one thing, book like Real analysis modern techniques written by Folland follows this definition. Actually this definition is an extended version of the definition of bounded variation in compact interval. Another fact is that, for a Lebesgue integrable function $f$ on real line the function $F(x)=int_{-infty}^x f(t)dt$ satisfies the condition given in definition, as we do for a Lebesgue integrable function in compact interval.
$endgroup$
– UserS
Dec 2 '18 at 5:59
add a comment |
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