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Estimate of the Degree of a Polynomial Approximation of log(x)
The Next CEO of Stack OverflowHow do I use Weierstrass Approximation Theorem?Application of Weierstrass approximation theoremapproximation of rational functionsApproximation of continuous functionsContinuity of the Stone-Weierstrass approximation operatorAre there guarantees on the $ L^2$ error of the best polynomial approximation (or pessimistic theorems)?How well can continuous functions on $[0,1]$ be approximated by polynomials up to a given degree?Showing the existence of a polynomial $p$ to approximate $f : [2,7] rightarrow BbbR$Uniform-degree polynomial estimation of smooth functions with bounded derivativesQuestion Using Weierstrass Approximation Thm:How do I use Weierstrass Approximation Theorem?
$begingroup$
I have a homework question asking me to find an estimate for the degree $n$ of a polynomial $P(x)$ that approximates $f(x) = log(x)$ on $[1,2]$ such that $suplimits_x in [1,2] |P(x) - log(x)| leq epsilon$ for $epsilon in [0,1]$.
From the Weierstrass approximation theorem (WAT) I know such a polynomial exists, and I am trying to tease something out following a proof using Bernstein polynomials because a hint says I can solve this problem by "evaluating the proof of the WAT."
I get lost trying to set up an inequality like $|B_n(f)(x) - f(x)| leq $ to start with, and I'm not sure what other way(s) I can approach this using/building off a proof of WAT.
As I understand the problem, since smaller and smaller $epsilon$ imply closer approximations to $f(x)$ on the given interval, certain thresholds of error necessitate the use of higher degree approximations. So given $epsilon in [0,1]$, I am asked to find an estimate of $n(epsilon)$ (not necessarily the best one) such that there always exists an approximation of $f(x) = log(x)$ by $P(x)$ with degree $n(epsilon)$ where $suplimits_x in [1,2] |P(x) - log(x)| leq epsilon$.
Provided that interpretation is correct, I found this similar question, but the given answer is not clicking for me and I can't apply the explanation because of that.
If anyone can lead me the right way or help me put some tools in my toolbox, I would appreciate it.
real-analysis analysis
edited Mar 27 at 20:30
Bernard
123k741117
asked Mar 27 at 20:19
kh7kh7
61
New contributor
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I have a homework question asking me to find an estimate for the degree $n$ of a polynomial $P(x)$ that approximates $f(x) = log(x)$ on $[1,2]$ such that $suplimits_x in [1,2] |P(x) - log(x)| leq epsilon$ for $epsilon in [0,1]$.
From the Weierstrass approximation theorem (WAT) I know such a polynomial exists, and I am trying to tease something out following a proof using Bernstein polynomials because a hint says I can solve this problem by "evaluating the proof of the WAT."
I get lost trying to set up an inequality like $|B_n(f)(x) - f(x)| leq $ to start with, and I'm not sure what other way(s) I can approach this using/building off a proof of WAT.
As I understand the problem, since smaller and smaller $epsilon$ imply closer approximations to $f(x)$ on the given interval, certain thresholds of error necessitate the use of higher degree approximations. So given $epsilon in [0,1]$, I am asked to find an estimate of $n(epsilon)$ (not necessarily the best one) such that there always exists an approximation of $f(x) = log(x)$ by $P(x)$ with degree $n(epsilon)$ where $suplimits_x in [1,2] |P(x) - log(x)| leq epsilon$.
Provided that interpretation is correct, I found this similar question, but the given answer is not clicking for me and I can't apply the explanation because of that.
If anyone can lead me the right way or help me put some tools in my toolbox, I would appreciate it.
real-analysis analysis
edited Mar 27 at 20:30
Bernard
123k741117
asked Mar 27 at 20:19
kh7kh7
61
New contributor
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I think your efforts would be well spent in trying to understand that previous Question's Answer. A basic step will be to discover what function $g(t)$ will work with your problem (to approximate $log x$ with Bernstein polynomials). I think it would be expeditious to help you if we can narrow your Question to helping you understand steps in such a previous analysis (as they apply to your problem).
$endgroup$
– hardmath
Mar 27 at 20:37
$begingroup$
In that answer, I am actually confused about the initial expression of $B_n(f)(x)$, as I haven't encountered it expressed with $f(frac2k -nn)$, etc. but rather, I've seen it in the form provided in the wiki link. Following that, I don't quite understand the introduction of $g(2)$ (I guess idk why 2, specifically) or how to find what $g(t)$ works for $log x$. The rest seems straight forward; although with how the question was posed, I was hoping there would be a way to avoid using pmf's, variance, etc. as I don't remember those topics being involved in our studies.
$endgroup$
– kh7
Mar 27 at 21:05
$begingroup$
Okay, that additional context is helpful to preparing an answer for you.
$endgroup$
– hardmath
Mar 27 at 21:16
add a comment |
$begingroup$
I have a homework question asking me to find an estimate for the degree $n$ of a polynomial $P(x)$ that approximates $f(x) = log(x)$ on $[1,2]$ such that $suplimits_x in [1,2] |P(x) - log(x)| leq epsilon$ for $epsilon in [0,1]$.
From the Weierstrass approximation theorem (WAT) I know such a polynomial exists, and I am trying to tease something out following a proof using Bernstein polynomials because a hint says I can solve this problem by "evaluating the proof of the WAT."
I get lost trying to set up an inequality like $|B_n(f)(x) - f(x)| leq $ to start with, and I'm not sure what other way(s) I can approach this using/building off a proof of WAT.
As I understand the problem, since smaller and smaller $epsilon$ imply closer approximations to $f(x)$ on the given interval, certain thresholds of error necessitate the use of higher degree approximations. So given $epsilon in [0,1]$, I am asked to find an estimate of $n(epsilon)$ (not necessarily the best one) such that there always exists an approximation of $f(x) = log(x)$ by $P(x)$ with degree $n(epsilon)$ where $suplimits_x in [1,2] |P(x) - log(x)| leq epsilon$.
Provided that interpretation is correct, I found this similar question, but the given answer is not clicking for me and I can't apply the explanation because of that.
If anyone can lead me the right way or help me put some tools in my toolbox, I would appreciate it.
real-analysis analysis
edited Mar 27 at 20:30
Bernard
123k741117
asked Mar 27 at 20:19
kh7kh7
61
New contributor
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I have a homework question asking me to find an estimate for the degree $n$ of a polynomial $P(x)$ that approximates $f(x) = log(x)$ on $[1,2]$ such that $suplimits_x in [1,2] |P(x) - log(x)| leq epsilon$ for $epsilon in [0,1]$.
From the Weierstrass approximation theorem (WAT) I know such a polynomial exists, and I am trying to tease something out following a proof using Bernstein polynomials because a hint says I can solve this problem by "evaluating the proof of the WAT."
I get lost trying to set up an inequality like $|B_n(f)(x) - f(x)| leq $ to start with, and I'm not sure what other way(s) I can approach this using/building off a proof of WAT.
As I understand the problem, since smaller and smaller $epsilon$ imply closer approximations to $f(x)$ on the given interval, certain thresholds of error necessitate the use of higher degree approximations. So given $epsilon in [0,1]$, I am asked to find an estimate of $n(epsilon)$ (not necessarily the best one) such that there always exists an approximation of $f(x) = log(x)$ by $P(x)$ with degree $n(epsilon)$ where $suplimits_x in [1,2] |P(x) - log(x)| leq epsilon$.
Provided that interpretation is correct, I found this similar question, but the given answer is not clicking for me and I can't apply the explanation because of that.
If anyone can lead me the right way or help me put some tools in my toolbox, I would appreciate it.
real-analysis analysis
real-analysis analysis
edited Mar 27 at 20:30
Bernard
123k741117
asked Mar 27 at 20:19
kh7kh7
61
New contributor
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 27 at 20:30
Bernard
123k741117
asked Mar 27 at 20:19
kh7kh7
61
New contributor
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 27 at 20:30
Bernard
123k741117
edited Mar 27 at 20:30
Bernard
123k741117
edited Mar 27 at 20:30
Bernard
123k741117
123k741117
asked Mar 27 at 20:19
kh7kh7
61
New contributor
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 27 at 20:19
kh7kh7
61
asked Mar 27 at 20:19
kh7kh7
61
61
New contributor
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
kh7 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
I think your efforts would be well spent in trying to understand that previous Question's Answer. A basic step will be to discover what function $g(t)$ will work with your problem (to approximate $log x$ with Bernstein polynomials). I think it would be expeditious to help you if we can narrow your Question to helping you understand steps in such a previous analysis (as they apply to your problem).
$endgroup$
– hardmath
Mar 27 at 20:37
$begingroup$
In that answer, I am actually confused about the initial expression of $B_n(f)(x)$, as I haven't encountered it expressed with $f(frac2k -nn)$, etc. but rather, I've seen it in the form provided in the wiki link. Following that, I don't quite understand the introduction of $g(2)$ (I guess idk why 2, specifically) or how to find what $g(t)$ works for $log x$. The rest seems straight forward; although with how the question was posed, I was hoping there would be a way to avoid using pmf's, variance, etc. as I don't remember those topics being involved in our studies.
$endgroup$
– kh7
Mar 27 at 21:05
$begingroup$
Okay, that additional context is helpful to preparing an answer for you.
$endgroup$
– hardmath
Mar 27 at 21:16
add a comment |
$begingroup$
I think your efforts would be well spent in trying to understand that previous Question's Answer. A basic step will be to discover what function $g(t)$ will work with your problem (to approximate $log x$ with Bernstein polynomials). I think it would be expeditious to help you if we can narrow your Question to helping you understand steps in such a previous analysis (as they apply to your problem).
$endgroup$
– hardmath
Mar 27 at 20:37
$begingroup$
In that answer, I am actually confused about the initial expression of $B_n(f)(x)$, as I haven't encountered it expressed with $f(frac2k -nn)$, etc. but rather, I've seen it in the form provided in the wiki link. Following that, I don't quite understand the introduction of $g(2)$ (I guess idk why 2, specifically) or how to find what $g(t)$ works for $log x$. The rest seems straight forward; although with how the question was posed, I was hoping there would be a way to avoid using pmf's, variance, etc. as I don't remember those topics being involved in our studies.
$endgroup$
– kh7
Mar 27 at 21:05
$begingroup$
Okay, that additional context is helpful to preparing an answer for you.
$endgroup$
– hardmath
Mar 27 at 21:16
$begingroup$
I think your efforts would be well spent in trying to understand that previous Question's Answer. A basic step will be to discover what function $g(t)$ will work with your problem (to approximate $log x$ with Bernstein polynomials). I think it would be expeditious to help you if we can narrow your Question to helping you understand steps in such a previous analysis (as they apply to your problem).
$endgroup$
– hardmath
Mar 27 at 20:37
$begingroup$
I think your efforts would be well spent in trying to understand that previous Question's Answer. A basic step will be to discover what function $g(t)$ will work with your problem (to approximate $log x$ with Bernstein polynomials). I think it would be expeditious to help you if we can narrow your Question to helping you understand steps in such a previous analysis (as they apply to your problem).
$endgroup$
– hardmath
Mar 27 at 20:37
$begingroup$
In that answer, I am actually confused about the initial expression of $B_n(f)(x)$, as I haven't encountered it expressed with $f(frac2k -nn)$, etc. but rather, I've seen it in the form provided in the wiki link. Following that, I don't quite understand the introduction of $g(2)$ (I guess idk why 2, specifically) or how to find what $g(t)$ works for $log x$. The rest seems straight forward; although with how the question was posed, I was hoping there would be a way to avoid using pmf's, variance, etc. as I don't remember those topics being involved in our studies.
$endgroup$
– kh7
Mar 27 at 21:05
$begingroup$
In that answer, I am actually confused about the initial expression of $B_n(f)(x)$, as I haven't encountered it expressed with $f(frac2k -nn)$, etc. but rather, I've seen it in the form provided in the wiki link. Following that, I don't quite understand the introduction of $g(2)$ (I guess idk why 2, specifically) or how to find what $g(t)$ works for $log x$. The rest seems straight forward; although with how the question was posed, I was hoping there would be a way to avoid using pmf's, variance, etc. as I don't remember those topics being involved in our studies.
$endgroup$
– kh7
Mar 27 at 21:05
$begingroup$
Okay, that additional context is helpful to preparing an answer for you.
$endgroup$
– hardmath
Mar 27 at 21:16
$begingroup$
Okay, that additional context is helpful to preparing an answer for you.
$endgroup$
– hardmath
Mar 27 at 21:16
add a comment |
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$begingroup$
I think your efforts would be well spent in trying to understand that previous Question's Answer. A basic step will be to discover what function $g(t)$ will work with your problem (to approximate $log x$ with Bernstein polynomials). I think it would be expeditious to help you if we can narrow your Question to helping you understand steps in such a previous analysis (as they apply to your problem).
$endgroup$
– hardmath
Mar 27 at 20:37
$begingroup$
In that answer, I am actually confused about the initial expression of $B_n(f)(x)$, as I haven't encountered it expressed with $f(frac2k -nn)$, etc. but rather, I've seen it in the form provided in the wiki link. Following that, I don't quite understand the introduction of $g(2)$ (I guess idk why 2, specifically) or how to find what $g(t)$ works for $log x$. The rest seems straight forward; although with how the question was posed, I was hoping there would be a way to avoid using pmf's, variance, etc. as I don't remember those topics being involved in our studies.
$endgroup$
– kh7
Mar 27 at 21:05
$begingroup$
Okay, that additional context is helpful to preparing an answer for you.
$endgroup$
– hardmath
Mar 27 at 21:16