Approximate C0-Funktion with C1-Funktions
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Let $I=[a,b]$ and $fin C(I).$ I want to show that there exists a $gin C^1(I),$ such that for any $varepsilon > 0$ $|f(x)-g(x)|<varepsilon$ for all $x in I.$ By Stone-Weierstrass approximation theorem the statement is true, but is there some easier way to prove this? (beginner's calculus)
Furthermore I want to show that there exists a monotonous $h in C^1([c,d])$, $a<c<d<b$ with $h(c)=f(c), h(d)=f(d)$ and $h^prime(c)=h^prime(d)=0$. It should be possible to construct a third-degree polynomial that satisfies all criterias (by having local extrema at $c,d$), but I assume there is a much simplier way to proof its existence?
Many thanks for any kind of help!
real-analysis continuity approximation
asked Jan 18 at 12:48
J. DoeJ. Doe
11
$endgroup$
add a comment |
$begingroup$
Let $I=[a,b]$ and $fin C(I).$ I want to show that there exists a $gin C^1(I),$ such that for any $varepsilon > 0$ $|f(x)-g(x)|<varepsilon$ for all $x in I.$ By Stone-Weierstrass approximation theorem the statement is true, but is there some easier way to prove this? (beginner's calculus)
Furthermore I want to show that there exists a monotonous $h in C^1([c,d])$, $a<c<d<b$ with $h(c)=f(c), h(d)=f(d)$ and $h^prime(c)=h^prime(d)=0$. It should be possible to construct a third-degree polynomial that satisfies all criterias (by having local extrema at $c,d$), but I assume there is a much simplier way to proof its existence?
Many thanks for any kind of help!
real-analysis continuity approximation
asked Jan 18 at 12:48
J. DoeJ. Doe
11
$endgroup$
$begingroup$
en.wikipedia.org/wiki/Mollifier?wprov=sfla1
$endgroup$
– ecrin
Jan 18 at 13:03
add a comment |
$begingroup$
Let $I=[a,b]$ and $fin C(I).$ I want to show that there exists a $gin C^1(I),$ such that for any $varepsilon > 0$ $|f(x)-g(x)|<varepsilon$ for all $x in I.$ By Stone-Weierstrass approximation theorem the statement is true, but is there some easier way to prove this? (beginner's calculus)
Furthermore I want to show that there exists a monotonous $h in C^1([c,d])$, $a<c<d<b$ with $h(c)=f(c), h(d)=f(d)$ and $h^prime(c)=h^prime(d)=0$. It should be possible to construct a third-degree polynomial that satisfies all criterias (by having local extrema at $c,d$), but I assume there is a much simplier way to proof its existence?
Many thanks for any kind of help!
real-analysis continuity approximation
asked Jan 18 at 12:48
J. DoeJ. Doe
11
$endgroup$
Let $I=[a,b]$ and $fin C(I).$ I want to show that there exists a $gin C^1(I),$ such that for any $varepsilon > 0$ $|f(x)-g(x)|<varepsilon$ for all $x in I.$ By Stone-Weierstrass approximation theorem the statement is true, but is there some easier way to prove this? (beginner's calculus)
Furthermore I want to show that there exists a monotonous $h in C^1([c,d])$, $a<c<d<b$ with $h(c)=f(c), h(d)=f(d)$ and $h^prime(c)=h^prime(d)=0$. It should be possible to construct a third-degree polynomial that satisfies all criterias (by having local extrema at $c,d$), but I assume there is a much simplier way to proof its existence?
Many thanks for any kind of help!
real-analysis continuity approximation
real-analysis continuity approximation
asked Jan 18 at 12:48
J. DoeJ. Doe
11
asked Jan 18 at 12:48
J. DoeJ. Doe
11
asked Jan 18 at 12:48
J. DoeJ. Doe
11
asked Jan 18 at 12:48
J. DoeJ. Doe
11
asked Jan 18 at 12:48
J. DoeJ. Doe
11
11
$begingroup$
en.wikipedia.org/wiki/Mollifier?wprov=sfla1
$endgroup$
– ecrin
Jan 18 at 13:03
add a comment |
$begingroup$
en.wikipedia.org/wiki/Mollifier?wprov=sfla1
$endgroup$
– ecrin
Jan 18 at 13:03
$begingroup$
en.wikipedia.org/wiki/Mollifier?wprov=sfla1
$endgroup$
– ecrin
Jan 18 at 13:03
$begingroup$
en.wikipedia.org/wiki/Mollifier?wprov=sfla1
$endgroup$
– ecrin
Jan 18 at 13:03
add a comment |
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$begingroup$
en.wikipedia.org/wiki/Mollifier?wprov=sfla1
$endgroup$
– ecrin
Jan 18 at 13:03