- #1
GregA
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SOLVED
With the assumption f(x)\in L^1(\mathbb{R})\cap L^2(\mathbb{R}) (in a Lebesque sense) I'm trying to include a short proof of Plancherel's identity into my dissertation but am having trouble justifying the change of integration at the end of the following line:
<br /> \int_{\mathbb{R}}|f(x)|^2\, \mathrm{d}x<br /> =\int_{\mathbb{R}}f(x)\overline{f(x)}\mathrm{d}x=\int_{\mathbb{R}}f(x)\left [\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\overline{{\hat{f}(\xi)e^{ix\xi}}}\mathrm{d}\xi\right ]\mathrm{d}x<br />
I am aware of Fubini's theorem but my problem is that I can't see why the integral of \left |\hat{f}(\xi)\right | is necessarily bounded in L^1 so that I can use it. [EDIT] Am I allowed to say that since I know f(x)\in L^2 then the equalitities give that \int f(\xi)\mathrm\,{d}\xi is bounded?
A number of texts just swap \mathrm{d}x and \mathrm{d}\xi without worrying this, whilst others just give a hand wavy argument that it all works out ok.
I know that it can be proved using the convolution theorem but the argument is longer. Am I missing something obvious or would anybody be able to point me in the right direction for resolving this problem? (and I acknowledge I have gaps in my knowledge about function spaces etc...)
With the assumption f(x)\in L^1(\mathbb{R})\cap L^2(\mathbb{R}) (in a Lebesque sense) I'm trying to include a short proof of Plancherel's identity into my dissertation but am having trouble justifying the change of integration at the end of the following line:
<br /> \int_{\mathbb{R}}|f(x)|^2\, \mathrm{d}x<br /> =\int_{\mathbb{R}}f(x)\overline{f(x)}\mathrm{d}x=\int_{\mathbb{R}}f(x)\left [\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\overline{{\hat{f}(\xi)e^{ix\xi}}}\mathrm{d}\xi\right ]\mathrm{d}x<br />
I am aware of Fubini's theorem but my problem is that I can't see why the integral of \left |\hat{f}(\xi)\right | is necessarily bounded in L^1 so that I can use it. [EDIT] Am I allowed to say that since I know f(x)\in L^2 then the equalitities give that \int f(\xi)\mathrm\,{d}\xi is bounded?
A number of texts just swap \mathrm{d}x and \mathrm{d}\xi without worrying this, whilst others just give a hand wavy argument that it all works out ok.
I know that it can be proved using the convolution theorem but the argument is longer. Am I missing something obvious or would anybody be able to point me in the right direction for resolving this problem? (and I acknowledge I have gaps in my knowledge about function spaces etc...)
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