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Does Wikipedia misstate Glasser's master theorem
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Citing from Wikipedia and one of its references, MathWorld, following hold:
Glasser's master Theorem For $f$ integrable, $Phi(x) = |a|x - sum_i=1^N fracalpha_ix-beta_i$ and $a$, $alpha_i$, $beta_i$ arbitrary real constants the identity
beginequation
mathrmPVint_-infty^infty f(Phi(x)) dx =
mathrmPV int_-infty^infty f(x) dx
labelGlasser
tag1
endequation
holds.
Now consider
beginalign*
Phi_1(x) &= |a|x - sum_i=1^N fracalpha_ix-beta_i \
Phi_2(x) &= x - sum_i=1^N fracbeta_i
endalign*
Then, by Glasser's theorem refGlasser
$$mathrmPVint_-infty^infty f(Phi_1(x)) dx =
mathrmPV int_-infty^infty f(x) dx =
mathrmPVint_-infty^infty f(Phi_2(x)).$$
However, under the change of variables $y = |a| x$
beginequation
mathrmPVint_-infty^infty f(Phi_1(x)) dx =
frac1mathrmPV int_-infty^infty f(Phi_2(y)) dy.
labelmy Idea
tag2
endequation
Thus, I assume Glasser's theorem only holds for $|a| = 1$; a quick numerical check seems to support Eq. refmy Idea. Is Wikipedia and MathWorld wrong about this?
integration proof-verification
asked Jul 26 at 14:16
manthano
1707
add a comment |Â
up vote
7
down vote
favorite
Citing from Wikipedia and one of its references, MathWorld, following hold:
Glasser's master Theorem For $f$ integrable, $Phi(x) = |a|x - sum_i=1^N fracalpha_ix-beta_i$ and $a$, $alpha_i$, $beta_i$ arbitrary real constants the identity
beginequation
mathrmPVint_-infty^infty f(Phi(x)) dx =
mathrmPV int_-infty^infty f(x) dx
labelGlasser
tag1
endequation
holds.
Now consider
beginalign*
Phi_1(x) &= |a|x - sum_i=1^N fracalpha_ix-beta_i \
Phi_2(x) &= x - sum_i=1^N fracbeta_i
endalign*
Then, by Glasser's theorem refGlasser
$$mathrmPVint_-infty^infty f(Phi_1(x)) dx =
mathrmPV int_-infty^infty f(x) dx =
mathrmPVint_-infty^infty f(Phi_2(x)).$$
However, under the change of variables $y = |a| x$
beginequation
mathrmPVint_-infty^infty f(Phi_1(x)) dx =
frac1mathrmPV int_-infty^infty f(Phi_2(y)) dy.
labelmy Idea
tag2
endequation
Thus, I assume Glasser's theorem only holds for $|a| = 1$; a quick numerical check seems to support Eq. refmy Idea. Is Wikipedia and MathWorld wrong about this?
integration proof-verification
asked Jul 26 at 14:16
manthano
1707
Good question! Maybe that’s a flaw, but I’ve seen this version of master theorem too many times that I cannot believe it is wrong.
– Szeto
Jul 26 at 15:11
sos440.blogspot.com/2017/01/glassers-master-theorem.html?m=1
– Bob
Jul 26 at 15:31
The paper of Glasser: jstor.org/stable/2007531?seq=1#page_scan_tab_contents
– Calvin Khor
Jul 27 at 14:06
add a comment |Â
up vote
7
down vote
favorite
up vote
7
down vote
favorite
Citing from Wikipedia and one of its references, MathWorld, following hold:
Glasser's master Theorem For $f$ integrable, $Phi(x) = |a|x - sum_i=1^N fracalpha_ix-beta_i$ and $a$, $alpha_i$, $beta_i$ arbitrary real constants the identity
beginequation
mathrmPVint_-infty^infty f(Phi(x)) dx =
mathrmPV int_-infty^infty f(x) dx
labelGlasser
tag1
endequation
holds.
Now consider
beginalign*
Phi_1(x) &= |a|x - sum_i=1^N fracalpha_ix-beta_i \
Phi_2(x) &= x - sum_i=1^N fracbeta_i
endalign*
Then, by Glasser's theorem refGlasser
$$mathrmPVint_-infty^infty f(Phi_1(x)) dx =
mathrmPV int_-infty^infty f(x) dx =
mathrmPVint_-infty^infty f(Phi_2(x)).$$
However, under the change of variables $y = |a| x$
beginequation
mathrmPVint_-infty^infty f(Phi_1(x)) dx =
frac1mathrmPV int_-infty^infty f(Phi_2(y)) dy.
labelmy Idea
tag2
endequation
Thus, I assume Glasser's theorem only holds for $|a| = 1$; a quick numerical check seems to support Eq. refmy Idea. Is Wikipedia and MathWorld wrong about this?
integration proof-verification
asked Jul 26 at 14:16
manthano
1707
Citing from Wikipedia and one of its references, MathWorld, following hold:
Glasser's master Theorem For $f$ integrable, $Phi(x) = |a|x - sum_i=1^N fracalpha_ix-beta_i$ and $a$, $alpha_i$, $beta_i$ arbitrary real constants the identity
beginequation
mathrmPVint_-infty^infty f(Phi(x)) dx =
mathrmPV int_-infty^infty f(x) dx
labelGlasser
tag1
endequation
holds.
Now consider
beginalign*
Phi_1(x) &= |a|x - sum_i=1^N fracalpha_ix-beta_i \
Phi_2(x) &= x - sum_i=1^N fracbeta_i
endalign*
Then, by Glasser's theorem refGlasser
$$mathrmPVint_-infty^infty f(Phi_1(x)) dx =
mathrmPV int_-infty^infty f(x) dx =
mathrmPVint_-infty^infty f(Phi_2(x)).$$
However, under the change of variables $y = |a| x$
beginequation
mathrmPVint_-infty^infty f(Phi_1(x)) dx =
frac1mathrmPV int_-infty^infty f(Phi_2(y)) dy.
labelmy Idea
tag2
endequation
Thus, I assume Glasser's theorem only holds for $|a| = 1$; a quick numerical check seems to support Eq. refmy Idea. Is Wikipedia and MathWorld wrong about this?
integration proof-verification
asked Jul 26 at 14:16
manthano
1707
edited Jul 26 at 15:06
asked Jul 26 at 14:16
manthano
1707
asked Jul 26 at 14:16
manthano
1707
asked Jul 26 at 14:16
manthano
1707
1707
Good question! Maybe that’s a flaw, but I’ve seen this version of master theorem too many times that I cannot believe it is wrong.
– Szeto
Jul 26 at 15:11
sos440.blogspot.com/2017/01/glassers-master-theorem.html?m=1
– Bob
Jul 26 at 15:31
The paper of Glasser: jstor.org/stable/2007531?seq=1#page_scan_tab_contents
– Calvin Khor
Jul 27 at 14:06
add a comment |Â
Good question! Maybe that’s a flaw, but I’ve seen this version of master theorem too many times that I cannot believe it is wrong.
– Szeto
Jul 26 at 15:11
sos440.blogspot.com/2017/01/glassers-master-theorem.html?m=1
– Bob
Jul 26 at 15:31
The paper of Glasser: jstor.org/stable/2007531?seq=1#page_scan_tab_contents
– Calvin Khor
Jul 27 at 14:06
Good question! Maybe that’s a flaw, but I’ve seen this version of master theorem too many times that I cannot believe it is wrong.
– Szeto
Jul 26 at 15:11
Good question! Maybe that’s a flaw, but I’ve seen this version of master theorem too many times that I cannot believe it is wrong.
– Szeto
Jul 26 at 15:11
sos440.blogspot.com/2017/01/glassers-master-theorem.html?m=1
– Bob
Jul 26 at 15:31
sos440.blogspot.com/2017/01/glassers-master-theorem.html?m=1
– Bob
Jul 26 at 15:31
The paper of Glasser: jstor.org/stable/2007531?seq=1#page_scan_tab_contents
– Calvin Khor
Jul 27 at 14:06
The paper of Glasser: jstor.org/stable/2007531?seq=1#page_scan_tab_contents
– Calvin Khor
Jul 27 at 14:06
add a comment |Â
1 Answer
1
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votes
up vote
1
down vote
Thank you for noticing this. Your argument is correct. (An alternative way to see the error is simply to set every $α_i$ to 0.) I fixed the Wikipedia page (it now uses $x-a$ in place of $|a|x$) and submitted a correction to MathWorld.
answered 2 days ago
Dmytro Taranovsky
416116
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Thank you for noticing this. Your argument is correct. (An alternative way to see the error is simply to set every $α_i$ to 0.) I fixed the Wikipedia page (it now uses $x-a$ in place of $|a|x$) and submitted a correction to MathWorld.
answered 2 days ago
Dmytro Taranovsky
416116
add a comment |Â
up vote
1
down vote
Thank you for noticing this. Your argument is correct. (An alternative way to see the error is simply to set every $α_i$ to 0.) I fixed the Wikipedia page (it now uses $x-a$ in place of $|a|x$) and submitted a correction to MathWorld.
answered 2 days ago
Dmytro Taranovsky
416116
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Thank you for noticing this. Your argument is correct. (An alternative way to see the error is simply to set every $α_i$ to 0.) I fixed the Wikipedia page (it now uses $x-a$ in place of $|a|x$) and submitted a correction to MathWorld.
answered 2 days ago
Dmytro Taranovsky
416116
Thank you for noticing this. Your argument is correct. (An alternative way to see the error is simply to set every $α_i$ to 0.) I fixed the Wikipedia page (it now uses $x-a$ in place of $|a|x$) and submitted a correction to MathWorld.
answered 2 days ago
Dmytro Taranovsky
416116
answered 2 days ago
Dmytro Taranovsky
416116
answered 2 days ago
Dmytro Taranovsky
416116
answered 2 days ago
Dmytro Taranovsky
416116
416116
add a comment |Â
add a comment |Â
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Good question! Maybe that’s a flaw, but I’ve seen this version of master theorem too many times that I cannot believe it is wrong.
– Szeto
Jul 26 at 15:11
sos440.blogspot.com/2017/01/glassers-master-theorem.html?m=1
– Bob
Jul 26 at 15:31
The paper of Glasser: jstor.org/stable/2007531?seq=1#page_scan_tab_contents
– Calvin Khor
Jul 27 at 14:06