Mixing
Approximation of an $L^1-$ function of two variables by trigonometric polynomials.
$begingroup$
We know as a theorem that the trigonometric polynomials are dense in $L^1([0,1))$
For instance for a Lebesgue integrable function we use the Fejer kernel $$F(x)=sum_{n=-N}^N(1-frac{|n|}{N+1})e^{2pi inx}$$ and we take $f*F(x)$
where $''*''$ denotes the convolution of two functions.
Is there an analogue of this theorem in two dimensions,i.e in $L^1([0,1)^2)$?
If there is,then where can i find o proof of this result?
Thank you in advance.
real-analysis fourier-analysis lebesgue-integral fourier-series approximation
asked Jan 19 at 13:09
Marios GretsasMarios Gretsas
8,48011437
$endgroup$
add a comment |
$begingroup$
We know as a theorem that the trigonometric polynomials are dense in $L^1([0,1))$
For instance for a Lebesgue integrable function we use the Fejer kernel $$F(x)=sum_{n=-N}^N(1-frac{|n|}{N+1})e^{2pi inx}$$ and we take $f*F(x)$
where $''*''$ denotes the convolution of two functions.
Is there an analogue of this theorem in two dimensions,i.e in $L^1([0,1)^2)$?
If there is,then where can i find o proof of this result?
Thank you in advance.
real-analysis fourier-analysis lebesgue-integral fourier-series approximation
asked Jan 19 at 13:09
Marios GretsasMarios Gretsas
8,48011437
$endgroup$
2
$begingroup$
Stone Weierstrass
$endgroup$
– mathworker21
Jan 19 at 13:11
1
$begingroup$
I think this would work: $$int_0^1int_0^1 f(x',y')F_N(x-x')F_M(y-y')dxdy$$.
$endgroup$
– DisintegratingByParts
Jan 19 at 23:03
add a comment |
$begingroup$
We know as a theorem that the trigonometric polynomials are dense in $L^1([0,1))$
For instance for a Lebesgue integrable function we use the Fejer kernel $$F(x)=sum_{n=-N}^N(1-frac{|n|}{N+1})e^{2pi inx}$$ and we take $f*F(x)$
where $''*''$ denotes the convolution of two functions.
Is there an analogue of this theorem in two dimensions,i.e in $L^1([0,1)^2)$?
If there is,then where can i find o proof of this result?
Thank you in advance.
real-analysis fourier-analysis lebesgue-integral fourier-series approximation
asked Jan 19 at 13:09
Marios GretsasMarios Gretsas
8,48011437
$endgroup$
We know as a theorem that the trigonometric polynomials are dense in $L^1([0,1))$
For instance for a Lebesgue integrable function we use the Fejer kernel $$F(x)=sum_{n=-N}^N(1-frac{|n|}{N+1})e^{2pi inx}$$ and we take $f*F(x)$
where $''*''$ denotes the convolution of two functions.
Is there an analogue of this theorem in two dimensions,i.e in $L^1([0,1)^2)$?
If there is,then where can i find o proof of this result?
Thank you in advance.
real-analysis fourier-analysis lebesgue-integral fourier-series approximation
real-analysis fourier-analysis lebesgue-integral fourier-series approximation
asked Jan 19 at 13:09
Marios GretsasMarios Gretsas
8,48011437
asked Jan 19 at 13:09
Marios GretsasMarios Gretsas
8,48011437
edited Jan 19 at 13:13
Marios Gretsas
asked Jan 19 at 13:09
Marios GretsasMarios Gretsas
8,48011437
asked Jan 19 at 13:09
Marios GretsasMarios Gretsas
8,48011437
asked Jan 19 at 13:09
Marios GretsasMarios Gretsas
8,48011437
8,48011437
2
$begingroup$
Stone Weierstrass
$endgroup$
– mathworker21
Jan 19 at 13:11
1
$begingroup$
I think this would work: $$int_0^1int_0^1 f(x',y')F_N(x-x')F_M(y-y')dxdy$$.
$endgroup$
– DisintegratingByParts
Jan 19 at 23:03
add a comment |
2
$begingroup$
Stone Weierstrass
$endgroup$
– mathworker21
Jan 19 at 13:11
1
$begingroup$
I think this would work: $$int_0^1int_0^1 f(x',y')F_N(x-x')F_M(y-y')dxdy$$.
$endgroup$
– DisintegratingByParts
Jan 19 at 23:03
2
2
$begingroup$
Stone Weierstrass
$endgroup$
– mathworker21
Jan 19 at 13:11
$begingroup$
Stone Weierstrass
$endgroup$
– mathworker21
Jan 19 at 13:11
1
1
$begingroup$
I think this would work: $$int_0^1int_0^1 f(x',y')F_N(x-x')F_M(y-y')dxdy$$.
$endgroup$
– DisintegratingByParts
Jan 19 at 23:03
$begingroup$
I think this would work: $$int_0^1int_0^1 f(x',y')F_N(x-x')F_M(y-y')dxdy$$.
$endgroup$
– DisintegratingByParts
Jan 19 at 23:03
add a comment |
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$begingroup$
Stone Weierstrass
$endgroup$
– mathworker21
Jan 19 at 13:11
1
$begingroup$
I think this would work: $$int_0^1int_0^1 f(x',y')F_N(x-x')F_M(y-y')dxdy$$.
$endgroup$
– DisintegratingByParts
Jan 19 at 23:03