When does “Gaussian integrability” imply regular integrability?
$begingroup$
Let $varphi:mathbb Rtomathbb C$, and suppose that the limit
$$lim_{sigmatoinfty}int_{-infty}^infty varphileft(xright)expleft{-frac12cdotleft(frac xsigmaright)^2right},dx$$
exists. Well, this limit converges intuitively to the integral of $varphi$. What conditions do we need to impose on $varphi$ so that the limit
$$lim_{sigmatoinfty}int_{-sigma}^sigma varphileft(xright),dx$$
also exists? Could this limit be different from the other one above with the Gaussian? I know that if $varphiin L^1$, then it's easy, but I can't find non-$L^1$ counterexamples.
The context is: I'm trying to derive the Fourier inversion formula for a specific class of functions. The textbook I'm using defines the space $mathcal M$ of "moderate decrease" functions as the class of functions $f:mathbb Rtomathbb C$ that are continuous and $O(1/x^2)$ at $pminfty$. I can show that, if $finmathcal M$, then the Fourier transform of $f$ is continuous and bounded (actually, not only bounded but goes to zero at infinity). However, even knowing those properties, when I try to prove the Fourier inversion formula, all I can show is that
$$f(0) = lim_{sigmatoinfty}int_{-infty}^infty varphileft(xiright)expleft{-frac12cdotleft(frac xisigmaright)^2right},dxi$$ where $varphi$ is the Fourier transform of $f$. Obviously, I actually wanted to show that: $$f(0) = lim_{sigmatoinfty}int_{-sigma}^sigma varphileft(xiright),dxi.$$
real-analysis integration fourier-analysis fourier-transform gaussian-integral
edited Dec 24 '18 at 22:09
Math1000
19.1k31745
asked Dec 24 '18 at 18:56
foninifonini
1,78911038
$endgroup$
add a comment |
$begingroup$
Let $varphi:mathbb Rtomathbb C$, and suppose that the limit
$$lim_{sigmatoinfty}int_{-infty}^infty varphileft(xright)expleft{-frac12cdotleft(frac xsigmaright)^2right},dx$$
exists. Well, this limit converges intuitively to the integral of $varphi$. What conditions do we need to impose on $varphi$ so that the limit
$$lim_{sigmatoinfty}int_{-sigma}^sigma varphileft(xright),dx$$
also exists? Could this limit be different from the other one above with the Gaussian? I know that if $varphiin L^1$, then it's easy, but I can't find non-$L^1$ counterexamples.
The context is: I'm trying to derive the Fourier inversion formula for a specific class of functions. The textbook I'm using defines the space $mathcal M$ of "moderate decrease" functions as the class of functions $f:mathbb Rtomathbb C$ that are continuous and $O(1/x^2)$ at $pminfty$. I can show that, if $finmathcal M$, then the Fourier transform of $f$ is continuous and bounded (actually, not only bounded but goes to zero at infinity). However, even knowing those properties, when I try to prove the Fourier inversion formula, all I can show is that
$$f(0) = lim_{sigmatoinfty}int_{-infty}^infty varphileft(xiright)expleft{-frac12cdotleft(frac xisigmaright)^2right},dxi$$ where $varphi$ is the Fourier transform of $f$. Obviously, I actually wanted to show that: $$f(0) = lim_{sigmatoinfty}int_{-sigma}^sigma varphileft(xiright),dxi.$$
real-analysis integration fourier-analysis fourier-transform gaussian-integral
edited Dec 24 '18 at 22:09
Math1000
19.1k31745
asked Dec 24 '18 at 18:56
foninifonini
1,78911038
$endgroup$
$begingroup$
What are the examples if $phi in L^1$?
$endgroup$
– zoidberg
Dec 24 '18 at 22:23
add a comment |
$begingroup$
Let $varphi:mathbb Rtomathbb C$, and suppose that the limit
$$lim_{sigmatoinfty}int_{-infty}^infty varphileft(xright)expleft{-frac12cdotleft(frac xsigmaright)^2right},dx$$
exists. Well, this limit converges intuitively to the integral of $varphi$. What conditions do we need to impose on $varphi$ so that the limit
$$lim_{sigmatoinfty}int_{-sigma}^sigma varphileft(xright),dx$$
also exists? Could this limit be different from the other one above with the Gaussian? I know that if $varphiin L^1$, then it's easy, but I can't find non-$L^1$ counterexamples.
The context is: I'm trying to derive the Fourier inversion formula for a specific class of functions. The textbook I'm using defines the space $mathcal M$ of "moderate decrease" functions as the class of functions $f:mathbb Rtomathbb C$ that are continuous and $O(1/x^2)$ at $pminfty$. I can show that, if $finmathcal M$, then the Fourier transform of $f$ is continuous and bounded (actually, not only bounded but goes to zero at infinity). However, even knowing those properties, when I try to prove the Fourier inversion formula, all I can show is that
$$f(0) = lim_{sigmatoinfty}int_{-infty}^infty varphileft(xiright)expleft{-frac12cdotleft(frac xisigmaright)^2right},dxi$$ where $varphi$ is the Fourier transform of $f$. Obviously, I actually wanted to show that: $$f(0) = lim_{sigmatoinfty}int_{-sigma}^sigma varphileft(xiright),dxi.$$
real-analysis integration fourier-analysis fourier-transform gaussian-integral
edited Dec 24 '18 at 22:09
Math1000
19.1k31745
asked Dec 24 '18 at 18:56
foninifonini
1,78911038
$endgroup$
Let $varphi:mathbb Rtomathbb C$, and suppose that the limit
$$lim_{sigmatoinfty}int_{-infty}^infty varphileft(xright)expleft{-frac12cdotleft(frac xsigmaright)^2right},dx$$
exists. Well, this limit converges intuitively to the integral of $varphi$. What conditions do we need to impose on $varphi$ so that the limit
$$lim_{sigmatoinfty}int_{-sigma}^sigma varphileft(xright),dx$$
also exists? Could this limit be different from the other one above with the Gaussian? I know that if $varphiin L^1$, then it's easy, but I can't find non-$L^1$ counterexamples.
The context is: I'm trying to derive the Fourier inversion formula for a specific class of functions. The textbook I'm using defines the space $mathcal M$ of "moderate decrease" functions as the class of functions $f:mathbb Rtomathbb C$ that are continuous and $O(1/x^2)$ at $pminfty$. I can show that, if $finmathcal M$, then the Fourier transform of $f$ is continuous and bounded (actually, not only bounded but goes to zero at infinity). However, even knowing those properties, when I try to prove the Fourier inversion formula, all I can show is that
$$f(0) = lim_{sigmatoinfty}int_{-infty}^infty varphileft(xiright)expleft{-frac12cdotleft(frac xisigmaright)^2right},dxi$$ where $varphi$ is the Fourier transform of $f$. Obviously, I actually wanted to show that: $$f(0) = lim_{sigmatoinfty}int_{-sigma}^sigma varphileft(xiright),dxi.$$
real-analysis integration fourier-analysis fourier-transform gaussian-integral
real-analysis integration fourier-analysis fourier-transform gaussian-integral
edited Dec 24 '18 at 22:09
Math1000
19.1k31745
asked Dec 24 '18 at 18:56
foninifonini
1,78911038
edited Dec 24 '18 at 22:09
Math1000
19.1k31745
asked Dec 24 '18 at 18:56
foninifonini
1,78911038
edited Dec 24 '18 at 22:09
Math1000
19.1k31745
edited Dec 24 '18 at 22:09
Math1000
19.1k31745
edited Dec 24 '18 at 22:09
Math1000
19.1k31745
19.1k31745
asked Dec 24 '18 at 18:56
foninifonini
1,78911038
asked Dec 24 '18 at 18:56
foninifonini
1,78911038
asked Dec 24 '18 at 18:56
foninifonini
1,78911038
1,78911038
$begingroup$
What are the examples if $phi in L^1$?
$endgroup$
– zoidberg
Dec 24 '18 at 22:23
add a comment |
$begingroup$
What are the examples if $phi in L^1$?
$endgroup$
– zoidberg
Dec 24 '18 at 22:23
$begingroup$
What are the examples if $phi in L^1$?
$endgroup$
– zoidberg
Dec 24 '18 at 22:23
$begingroup$
What are the examples if $phi in L^1$?
$endgroup$
– zoidberg
Dec 24 '18 at 22:23
add a comment |
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$begingroup$
What are the examples if $phi in L^1$?
$endgroup$
– zoidberg
Dec 24 '18 at 22:23