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 at 5:07
user398843
562215
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up vote
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down vote
favorite
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 at 5:07
user398843
562215
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 at 5:07
user398843
562215
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 at 5:07
user398843
562215
asked Dec 2 at 5:07
user398843
562215
asked Dec 2 at 5:07
user398843
562215
asked Dec 2 at 5:07
user398843
562215
asked Dec 2 at 5:07
user398843
562215
562215
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 at 5:16
UserS
1,441112
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.
– MathematicsStudent1122
Dec 2 at 5:21
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.
– UserS
Dec 2 at 5:59
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
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 at 5:16
UserS
1,441112
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.
– MathematicsStudent1122
Dec 2 at 5:21
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.
– UserS
Dec 2 at 5:59
add a comment |
up vote
0
down vote
accepted
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 at 5:16
UserS
1,441112
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.
– MathematicsStudent1122
Dec 2 at 5:21
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.
– UserS
Dec 2 at 5:59
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
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 at 5:16
UserS
1,441112
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 at 5:16
UserS
1,441112
answered Dec 2 at 5:16
UserS
1,441112
answered Dec 2 at 5:16
UserS
1,441112
answered Dec 2 at 5:16
UserS
1,441112
1,441112
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.
– MathematicsStudent1122
Dec 2 at 5:21
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.
– UserS
Dec 2 at 5:59
add a comment |
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.
– MathematicsStudent1122
Dec 2 at 5:21
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.
– UserS
Dec 2 at 5:59
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.
– MathematicsStudent1122
Dec 2 at 5:21
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.
– MathematicsStudent1122
Dec 2 at 5:21
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.
– UserS
Dec 2 at 5:59
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.
– UserS
Dec 2 at 5:59
add a comment |
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