Specific proof technique of the complex Stone-Weierstrass theorem
The question is as follows:
If $f:mathbb{T}rightarrowmathbb{C}$ is continuous, prove that there is a sequence of polynomials $p_n(z,bar{z})$ such that $p_nrightarrow f$ uniformly for every $zinmathbb{T}$.
(Note: $mathbb{T}$ denotes the unit circle.) I've seen proofs of the more general statement of the complex version (https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Stone%E2%80%93Weierstrass_theorem,_complex_version), but this is asked in the context of a first course in complex analysis, so we have not developed the foundation to even understand the more general statement. We are given the following hint, however.
Let $g(re^{itheta})=P_r(f)$ and show that for each $r<1$ there is a sequence of polynomials $p_n(z,bar{z})$ such that $p_n$ converge uniformly for every $zinmathbb{T}$.
(Note: $P_r(f)$ denotes the Poisson kernel.) Here are my specific questions:
1) Can we prove this by simply writing $f$ as $f=u+iv$ for some real-valued, continuous functions $u$ and $v$ and then applying the real version of Stone-Weierstrass? I.e. approximating $u$ and $v$ with polynomials of real variables and claiming that the supremum norm of $f$ minus the sum of these polynomials is arbitrarily small? (Applying the fact that polynomials in 2 real variables can be transformed into polynomials in complex conjugates of 1 variable.)
2) If the above is an invalid approach, how does introducing the Poisson kernel fix the logical error (as what I'm proposing is a similar idea to the hint)?
It is quite possible that I just have a fundamental misunderstanding of the Poisson kernel. Maybe my claim in 1) that polynomials in 2 real variables can be transformed into polynomials in complex conjugates of 1 variable is dependent on the Poisson kernel?
The purpose of this post is to request assistance in interpreting this problem (and required tools to prove it), not to ask for a solution.
real-analysis complex-analysis uniform-convergence weierstrass-approximation
asked Nov 24 at 21:07
Atsina
791116
add a comment |
The question is as follows:
If $f:mathbb{T}rightarrowmathbb{C}$ is continuous, prove that there is a sequence of polynomials $p_n(z,bar{z})$ such that $p_nrightarrow f$ uniformly for every $zinmathbb{T}$.
(Note: $mathbb{T}$ denotes the unit circle.) I've seen proofs of the more general statement of the complex version (https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Stone%E2%80%93Weierstrass_theorem,_complex_version), but this is asked in the context of a first course in complex analysis, so we have not developed the foundation to even understand the more general statement. We are given the following hint, however.
Let $g(re^{itheta})=P_r(f)$ and show that for each $r<1$ there is a sequence of polynomials $p_n(z,bar{z})$ such that $p_n$ converge uniformly for every $zinmathbb{T}$.
(Note: $P_r(f)$ denotes the Poisson kernel.) Here are my specific questions:
1) Can we prove this by simply writing $f$ as $f=u+iv$ for some real-valued, continuous functions $u$ and $v$ and then applying the real version of Stone-Weierstrass? I.e. approximating $u$ and $v$ with polynomials of real variables and claiming that the supremum norm of $f$ minus the sum of these polynomials is arbitrarily small? (Applying the fact that polynomials in 2 real variables can be transformed into polynomials in complex conjugates of 1 variable.)
2) If the above is an invalid approach, how does introducing the Poisson kernel fix the logical error (as what I'm proposing is a similar idea to the hint)?
It is quite possible that I just have a fundamental misunderstanding of the Poisson kernel. Maybe my claim in 1) that polynomials in 2 real variables can be transformed into polynomials in complex conjugates of 1 variable is dependent on the Poisson kernel?
The purpose of this post is to request assistance in interpreting this problem (and required tools to prove it), not to ask for a solution.
real-analysis complex-analysis uniform-convergence weierstrass-approximation
asked Nov 24 at 21:07
Atsina
791116
$p_nrightarrow f$ uniformly for every $zinmathbb{T}$. that doesn't really make sense.
– zhw.
Nov 25 at 1:13
@zhw. ...why not?
– Atsina
Nov 25 at 1:15
@Atsina what is your definition of convergence in $mathbb{C}?$
– Idonknow
Nov 25 at 3:43
@Idonknow A sequence of complex numbers ${z_1,z_2,cdots}$ converges to $winmathbb{C}$ if $lim_{nrightarrowinfty}|z_n-w|=0$? I mean, there are plenty of equivalent definitions and theorems involving them, but this is probably the simplest definition of convergence in $mathbb{C}$
– Atsina
Nov 25 at 3:51
add a comment |
The question is as follows:
If $f:mathbb{T}rightarrowmathbb{C}$ is continuous, prove that there is a sequence of polynomials $p_n(z,bar{z})$ such that $p_nrightarrow f$ uniformly for every $zinmathbb{T}$.
(Note: $mathbb{T}$ denotes the unit circle.) I've seen proofs of the more general statement of the complex version (https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Stone%E2%80%93Weierstrass_theorem,_complex_version), but this is asked in the context of a first course in complex analysis, so we have not developed the foundation to even understand the more general statement. We are given the following hint, however.
Let $g(re^{itheta})=P_r(f)$ and show that for each $r<1$ there is a sequence of polynomials $p_n(z,bar{z})$ such that $p_n$ converge uniformly for every $zinmathbb{T}$.
(Note: $P_r(f)$ denotes the Poisson kernel.) Here are my specific questions:
1) Can we prove this by simply writing $f$ as $f=u+iv$ for some real-valued, continuous functions $u$ and $v$ and then applying the real version of Stone-Weierstrass? I.e. approximating $u$ and $v$ with polynomials of real variables and claiming that the supremum norm of $f$ minus the sum of these polynomials is arbitrarily small? (Applying the fact that polynomials in 2 real variables can be transformed into polynomials in complex conjugates of 1 variable.)
2) If the above is an invalid approach, how does introducing the Poisson kernel fix the logical error (as what I'm proposing is a similar idea to the hint)?
It is quite possible that I just have a fundamental misunderstanding of the Poisson kernel. Maybe my claim in 1) that polynomials in 2 real variables can be transformed into polynomials in complex conjugates of 1 variable is dependent on the Poisson kernel?
The purpose of this post is to request assistance in interpreting this problem (and required tools to prove it), not to ask for a solution.
real-analysis complex-analysis uniform-convergence weierstrass-approximation
asked Nov 24 at 21:07
Atsina
791116
The question is as follows:
If $f:mathbb{T}rightarrowmathbb{C}$ is continuous, prove that there is a sequence of polynomials $p_n(z,bar{z})$ such that $p_nrightarrow f$ uniformly for every $zinmathbb{T}$.
(Note: $mathbb{T}$ denotes the unit circle.) I've seen proofs of the more general statement of the complex version (https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Stone%E2%80%93Weierstrass_theorem,_complex_version), but this is asked in the context of a first course in complex analysis, so we have not developed the foundation to even understand the more general statement. We are given the following hint, however.
Let $g(re^{itheta})=P_r(f)$ and show that for each $r<1$ there is a sequence of polynomials $p_n(z,bar{z})$ such that $p_n$ converge uniformly for every $zinmathbb{T}$.
(Note: $P_r(f)$ denotes the Poisson kernel.) Here are my specific questions:
1) Can we prove this by simply writing $f$ as $f=u+iv$ for some real-valued, continuous functions $u$ and $v$ and then applying the real version of Stone-Weierstrass? I.e. approximating $u$ and $v$ with polynomials of real variables and claiming that the supremum norm of $f$ minus the sum of these polynomials is arbitrarily small? (Applying the fact that polynomials in 2 real variables can be transformed into polynomials in complex conjugates of 1 variable.)
2) If the above is an invalid approach, how does introducing the Poisson kernel fix the logical error (as what I'm proposing is a similar idea to the hint)?
It is quite possible that I just have a fundamental misunderstanding of the Poisson kernel. Maybe my claim in 1) that polynomials in 2 real variables can be transformed into polynomials in complex conjugates of 1 variable is dependent on the Poisson kernel?
The purpose of this post is to request assistance in interpreting this problem (and required tools to prove it), not to ask for a solution.
real-analysis complex-analysis uniform-convergence weierstrass-approximation
real-analysis complex-analysis uniform-convergence weierstrass-approximation
asked Nov 24 at 21:07
Atsina
791116
asked Nov 24 at 21:07
Atsina
791116
asked Nov 24 at 21:07
Atsina
791116
asked Nov 24 at 21:07
Atsina
791116
asked Nov 24 at 21:07
Atsina
791116
791116
$p_nrightarrow f$ uniformly for every $zinmathbb{T}$. that doesn't really make sense.
– zhw.
Nov 25 at 1:13
@zhw. ...why not?
– Atsina
Nov 25 at 1:15
@Atsina what is your definition of convergence in $mathbb{C}?$
– Idonknow
Nov 25 at 3:43
@Idonknow A sequence of complex numbers ${z_1,z_2,cdots}$ converges to $winmathbb{C}$ if $lim_{nrightarrowinfty}|z_n-w|=0$? I mean, there are plenty of equivalent definitions and theorems involving them, but this is probably the simplest definition of convergence in $mathbb{C}$
– Atsina
Nov 25 at 3:51
add a comment |
$p_nrightarrow f$ uniformly for every $zinmathbb{T}$. that doesn't really make sense.
– zhw.
Nov 25 at 1:13
@zhw. ...why not?
– Atsina
Nov 25 at 1:15
@Atsina what is your definition of convergence in $mathbb{C}?$
– Idonknow
Nov 25 at 3:43
@Idonknow A sequence of complex numbers ${z_1,z_2,cdots}$ converges to $winmathbb{C}$ if $lim_{nrightarrowinfty}|z_n-w|=0$? I mean, there are plenty of equivalent definitions and theorems involving them, but this is probably the simplest definition of convergence in $mathbb{C}$
– Atsina
Nov 25 at 3:51
$p_nrightarrow f$ uniformly for every $zinmathbb{T}$. that doesn't really make sense.
– zhw.
Nov 25 at 1:13
$p_nrightarrow f$ uniformly for every $zinmathbb{T}$. that doesn't really make sense.
– zhw.
Nov 25 at 1:13
@zhw. ...why not?
– Atsina
Nov 25 at 1:15
@zhw. ...why not?
– Atsina
Nov 25 at 1:15
@Atsina what is your definition of convergence in $mathbb{C}?$
– Idonknow
Nov 25 at 3:43
@Atsina what is your definition of convergence in $mathbb{C}?$
– Idonknow
Nov 25 at 3:43
@Idonknow A sequence of complex numbers ${z_1,z_2,cdots}$ converges to $winmathbb{C}$ if $lim_{nrightarrowinfty}|z_n-w|=0$? I mean, there are plenty of equivalent definitions and theorems involving them, but this is probably the simplest definition of convergence in $mathbb{C}$
– Atsina
Nov 25 at 3:51
@Idonknow A sequence of complex numbers ${z_1,z_2,cdots}$ converges to $winmathbb{C}$ if $lim_{nrightarrowinfty}|z_n-w|=0$? I mean, there are plenty of equivalent definitions and theorems involving them, but this is probably the simplest definition of convergence in $mathbb{C}$
– Atsina
Nov 25 at 3:51
add a comment |
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$p_nrightarrow f$ uniformly for every $zinmathbb{T}$. that doesn't really make sense.
– zhw.
Nov 25 at 1:13
@zhw. ...why not?
– Atsina
Nov 25 at 1:15
@Atsina what is your definition of convergence in $mathbb{C}?$
– Idonknow
Nov 25 at 3:43
@Idonknow A sequence of complex numbers ${z_1,z_2,cdots}$ converges to $winmathbb{C}$ if $lim_{nrightarrowinfty}|z_n-w|=0$? I mean, there are plenty of equivalent definitions and theorems involving them, but this is probably the simplest definition of convergence in $mathbb{C}$
– Atsina
Nov 25 at 3:51