List of proofs of Weierstrass Approximation Theorem
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I am looking for different proofs of the theorem :
If $f$ is a continuous real-valued function on $[a,b]$ and if any $epsilon>0$ is given, then there exists a polynomial $p$ on $[a,b]$ such that
$|f(x)-P(x)|< epsilon $
for all $x in [a,b]$.
analysis
asked Dec 1 '18 at 20:02
mike mokemike moke
406
$endgroup$
add a comment |
$begingroup$
I am looking for different proofs of the theorem :
If $f$ is a continuous real-valued function on $[a,b]$ and if any $epsilon>0$ is given, then there exists a polynomial $p$ on $[a,b]$ such that
$|f(x)-P(x)|< epsilon $
for all $x in [a,b]$.
analysis
asked Dec 1 '18 at 20:02
mike mokemike moke
406
$endgroup$
add a comment |
$begingroup$
I am looking for different proofs of the theorem :
If $f$ is a continuous real-valued function on $[a,b]$ and if any $epsilon>0$ is given, then there exists a polynomial $p$ on $[a,b]$ such that
$|f(x)-P(x)|< epsilon $
for all $x in [a,b]$.
analysis
asked Dec 1 '18 at 20:02
mike mokemike moke
406
$endgroup$
I am looking for different proofs of the theorem :
If $f$ is a continuous real-valued function on $[a,b]$ and if any $epsilon>0$ is given, then there exists a polynomial $p$ on $[a,b]$ such that
$|f(x)-P(x)|< epsilon $
for all $x in [a,b]$.
analysis
analysis
asked Dec 1 '18 at 20:02
mike mokemike moke
406
asked Dec 1 '18 at 20:02
mike mokemike moke
406
asked Dec 1 '18 at 20:02
mike mokemike moke
406
asked Dec 1 '18 at 20:02
mike mokemike moke
406
asked Dec 1 '18 at 20:02
mike mokemike moke
406
406
add a comment |
add a comment |
1 Answer
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My favorite proof uses probability! Here are two exercises that will help you prove it.
Step 1: Let $Z_n$ be a sequence of random variables and $c$ a constant such that for each $epsilon > 0$ it holds
that $$mathbb{P}[|Z_n −c| > epsilon] rightarrow 0, text{as } n rightarrow 0.$$ Show that for any bounded continuous function $g,$
$$mathbb{E}[g(Z_n)] rightarrow g(c) text{as } n rightarrow 0.$$
A hint for Step $1$ is to use the fact that $g$ is bounded and the definition of continuity.
Step 2: Let $f(x)$ be a continuous function in $[0,1]$. Consider the $textit{Bernstein Polynomials,}$ defined by
$$ B_n(x) = sum_{k=0}^n f left( frac{k}n right) dbinom{n}k x^k (1-x)^{n-k}. $$
Show that $B_n rightarrow f$ uniformly.
A hint for Step $2$ is to use the weak law of large numbers.
answered Dec 1 '18 at 20:38
Sandeep SilwalSandeep Silwal
5,86811237
$endgroup$
$begingroup$
I have seen the proof which uses only Bernstein Polynomials . I am looking for a proof that uses Lagrange Interpolating Polynomial . Thanks for your post , I will try to prove it.
$endgroup$
– mike moke
Dec 2 '18 at 6:22
add a comment |
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1 Answer
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$begingroup$
My favorite proof uses probability! Here are two exercises that will help you prove it.
Step 1: Let $Z_n$ be a sequence of random variables and $c$ a constant such that for each $epsilon > 0$ it holds
that $$mathbb{P}[|Z_n −c| > epsilon] rightarrow 0, text{as } n rightarrow 0.$$ Show that for any bounded continuous function $g,$
$$mathbb{E}[g(Z_n)] rightarrow g(c) text{as } n rightarrow 0.$$
A hint for Step $1$ is to use the fact that $g$ is bounded and the definition of continuity.
Step 2: Let $f(x)$ be a continuous function in $[0,1]$. Consider the $textit{Bernstein Polynomials,}$ defined by
$$ B_n(x) = sum_{k=0}^n f left( frac{k}n right) dbinom{n}k x^k (1-x)^{n-k}. $$
Show that $B_n rightarrow f$ uniformly.
A hint for Step $2$ is to use the weak law of large numbers.
answered Dec 1 '18 at 20:38
Sandeep SilwalSandeep Silwal
5,86811237
$endgroup$
$begingroup$
I have seen the proof which uses only Bernstein Polynomials . I am looking for a proof that uses Lagrange Interpolating Polynomial . Thanks for your post , I will try to prove it.
$endgroup$
– mike moke
Dec 2 '18 at 6:22
add a comment |
$begingroup$
My favorite proof uses probability! Here are two exercises that will help you prove it.
Step 1: Let $Z_n$ be a sequence of random variables and $c$ a constant such that for each $epsilon > 0$ it holds
that $$mathbb{P}[|Z_n −c| > epsilon] rightarrow 0, text{as } n rightarrow 0.$$ Show that for any bounded continuous function $g,$
$$mathbb{E}[g(Z_n)] rightarrow g(c) text{as } n rightarrow 0.$$
A hint for Step $1$ is to use the fact that $g$ is bounded and the definition of continuity.
Step 2: Let $f(x)$ be a continuous function in $[0,1]$. Consider the $textit{Bernstein Polynomials,}$ defined by
$$ B_n(x) = sum_{k=0}^n f left( frac{k}n right) dbinom{n}k x^k (1-x)^{n-k}. $$
Show that $B_n rightarrow f$ uniformly.
A hint for Step $2$ is to use the weak law of large numbers.
answered Dec 1 '18 at 20:38
Sandeep SilwalSandeep Silwal
5,86811237
$endgroup$
$begingroup$
I have seen the proof which uses only Bernstein Polynomials . I am looking for a proof that uses Lagrange Interpolating Polynomial . Thanks for your post , I will try to prove it.
$endgroup$
– mike moke
Dec 2 '18 at 6:22
add a comment |
$begingroup$
My favorite proof uses probability! Here are two exercises that will help you prove it.
Step 1: Let $Z_n$ be a sequence of random variables and $c$ a constant such that for each $epsilon > 0$ it holds
that $$mathbb{P}[|Z_n −c| > epsilon] rightarrow 0, text{as } n rightarrow 0.$$ Show that for any bounded continuous function $g,$
$$mathbb{E}[g(Z_n)] rightarrow g(c) text{as } n rightarrow 0.$$
A hint for Step $1$ is to use the fact that $g$ is bounded and the definition of continuity.
Step 2: Let $f(x)$ be a continuous function in $[0,1]$. Consider the $textit{Bernstein Polynomials,}$ defined by
$$ B_n(x) = sum_{k=0}^n f left( frac{k}n right) dbinom{n}k x^k (1-x)^{n-k}. $$
Show that $B_n rightarrow f$ uniformly.
A hint for Step $2$ is to use the weak law of large numbers.
answered Dec 1 '18 at 20:38
Sandeep SilwalSandeep Silwal
5,86811237
$endgroup$
My favorite proof uses probability! Here are two exercises that will help you prove it.
Step 1: Let $Z_n$ be a sequence of random variables and $c$ a constant such that for each $epsilon > 0$ it holds
that $$mathbb{P}[|Z_n −c| > epsilon] rightarrow 0, text{as } n rightarrow 0.$$ Show that for any bounded continuous function $g,$
$$mathbb{E}[g(Z_n)] rightarrow g(c) text{as } n rightarrow 0.$$
A hint for Step $1$ is to use the fact that $g$ is bounded and the definition of continuity.
Step 2: Let $f(x)$ be a continuous function in $[0,1]$. Consider the $textit{Bernstein Polynomials,}$ defined by
$$ B_n(x) = sum_{k=0}^n f left( frac{k}n right) dbinom{n}k x^k (1-x)^{n-k}. $$
Show that $B_n rightarrow f$ uniformly.
A hint for Step $2$ is to use the weak law of large numbers.
answered Dec 1 '18 at 20:38
Sandeep SilwalSandeep Silwal
5,86811237
answered Dec 1 '18 at 20:38
Sandeep SilwalSandeep Silwal
5,86811237
answered Dec 1 '18 at 20:38
Sandeep SilwalSandeep Silwal
5,86811237
answered Dec 1 '18 at 20:38
Sandeep SilwalSandeep Silwal
5,86811237
5,86811237
$begingroup$
I have seen the proof which uses only Bernstein Polynomials . I am looking for a proof that uses Lagrange Interpolating Polynomial . Thanks for your post , I will try to prove it.
$endgroup$
– mike moke
Dec 2 '18 at 6:22
add a comment |
$begingroup$
I have seen the proof which uses only Bernstein Polynomials . I am looking for a proof that uses Lagrange Interpolating Polynomial . Thanks for your post , I will try to prove it.
$endgroup$
– mike moke
Dec 2 '18 at 6:22
$begingroup$
I have seen the proof which uses only Bernstein Polynomials . I am looking for a proof that uses Lagrange Interpolating Polynomial . Thanks for your post , I will try to prove it.
$endgroup$
– mike moke
Dec 2 '18 at 6:22
$begingroup$
I have seen the proof which uses only Bernstein Polynomials . I am looking for a proof that uses Lagrange Interpolating Polynomial . Thanks for your post , I will try to prove it.
$endgroup$
– mike moke
Dec 2 '18 at 6:22
add a comment |
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