- #1
GregA
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SOLVED
With the assumption [tex]f(x)\in L^1(\mathbb{R})\cap L^2(\mathbb{R})[/tex] (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:
[tex]
\int_{\mathbb{R}}|f(x)|^2\, \mathrm{d}x
=\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
[/tex]
I am aware of Fubini's theorem but my problem is that I can't see why the integral of [itex]\left |\hat{f}(\xi)\right |[/itex] is necessarily bounded in [tex]L^1[/tex] so that I can use it. [EDIT] Am I allowed to say that since I know [tex]f(x)\in L^2[/tex] then the equalitities give that [itex]\int f(\xi)\mathrm\,{d}\xi[/itex] is bounded?
A number of texts just swap [tex]\mathrm{d}x[/tex] and [tex]\mathrm{d}\xi[/tex] 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 [tex]f(x)\in L^1(\mathbb{R})\cap L^2(\mathbb{R})[/tex] (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:
[tex]
\int_{\mathbb{R}}|f(x)|^2\, \mathrm{d}x
=\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
[/tex]
I am aware of Fubini's theorem but my problem is that I can't see why the integral of [itex]\left |\hat{f}(\xi)\right |[/itex] is necessarily bounded in [tex]L^1[/tex] so that I can use it. [EDIT] Am I allowed to say that since I know [tex]f(x)\in L^2[/tex] then the equalitities give that [itex]\int f(\xi)\mathrm\,{d}\xi[/itex] is bounded?
A number of texts just swap [tex]\mathrm{d}x[/tex] and [tex]\mathrm{d}\xi[/tex] 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...)
Last edited: