diff options
Diffstat (limited to 'source/know/concept/parsevals-theorem')
| -rw-r--r-- | source/know/concept/parsevals-theorem/index.md | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/source/know/concept/parsevals-theorem/index.md b/source/know/concept/parsevals-theorem/index.md index 41e8fed..a7ce0bf 100644 --- a/source/know/concept/parsevals-theorem/index.md +++ b/source/know/concept/parsevals-theorem/index.md @@ -17,7 +17,7 @@ where $$A$$, $$B$$, and $$s$$ are constants from the FT's definition: $$\begin{aligned} \boxed{ \begin{aligned} - \Inprod{f(x)}{g(x)} &= \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}(k)}{\tilde{g}(k)} + \inprod{f(x)}{g(x)} &= \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}(k)}{\tilde{g}(k)} \\ \inprod{\tilde{f}(k)}{\tilde{g}(k)} &= \frac{2 \pi A^2}{|s|} \Inprod{f(x)}{g(x)} \end{aligned} @@ -29,7 +29,7 @@ $$\begin{aligned} We insert the inverse FT into the definition of the inner product: $$\begin{aligned} - \Inprod{f}{g} + \inprod{f}{g} &= \int_{-\infty}^\infty \big( \hat{\mathcal{F}}^{-1}\{\tilde{f}(k)\}\big)^* \: \hat{\mathcal{F}}^{-1}\{\tilde{g}(k)\} \dd{x} \\ &= B^2 \int @@ -65,7 +65,7 @@ $$\begin{aligned} &= 2 \pi A^2 \iint f^*(x') \: g(x) \: \delta\big(s (x \!-\! x')\big) \dd{x'} \dd{x} \\ &= \frac{2 \pi A^2}{|s|} \int_{-\infty}^\infty f^*(x) \: g(x) \dd{x} - = \frac{2 \pi A^2}{|s|} \Inprod{f}{g} + = \frac{2 \pi A^2}{|s|} \inprod{f}{g} \end{aligned}$$ {% include proof/end.html id="proof-fourier" %} |