diff options
Diffstat (limited to 'source/know/concept/parsevals-theorem')
| -rw-r--r-- | source/know/concept/parsevals-theorem/index.md | 20 |
1 files changed, 11 insertions, 9 deletions
diff --git a/source/know/concept/parsevals-theorem/index.md b/source/know/concept/parsevals-theorem/index.md index 377f3a1..41e8fed 100644 --- a/source/know/concept/parsevals-theorem/index.md +++ b/source/know/concept/parsevals-theorem/index.md @@ -26,20 +26,21 @@ $$\begin{aligned} {% include proof/start.html id="proof-fourier" -%} -We insert the inverse FT into the defintion of the inner product: +We insert the inverse FT into the definition of the inner product: $$\begin{aligned} \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 - \Big( \int \tilde{f}^*(k_1) \exp(i s k_1 x) \dd{k_1} \Big) - \Big( \int \tilde{g}(k) \exp(- i s k x) \dd{k} \Big) + \Big( \int \tilde{f}^*(k') \: e^{i s k' x} \dd{k'} \Big) + \Big( \int \tilde{g}(k) \: e^{- i s k x} \dd{k} \Big) \dd{x} \\ - &= 2 \pi B^2 \iint \tilde{f}^*(k_1) \tilde{g}(k) \Big( \frac{1}{2 \pi} \int_{-\infty}^\infty \exp(i s x (k_1 - k)) \dd{x} \Big) \dd{k_1} \dd{k} + &= 2 \pi B^2 \iint \tilde{f}^*(k') \: \tilde{g}(k) \Big( \frac{1}{2 \pi} + \int_{-\infty}^\infty e^{i s x (k' - k)} \dd{x} \Big) \dd{k'} \dd{k} \\ - &= 2 \pi B^2 \iint \tilde{f}^*(k_1) \: \tilde{g}(k) \: \delta(s (k_1 - k)) \dd{k_1} \dd{k} + &= 2 \pi B^2 \iint \tilde{f}^*(k') \: \tilde{g}(k) \: \delta\big(s (k' \!-\! k)\big) \dd{k'} \dd{k} \\ &= \frac{2 \pi B^2}{|s|} \int_{-\infty}^\infty \tilde{f}^*(k) \: \tilde{g}(k) \dd{k} = \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}}{\tilde{g}} @@ -54,13 +55,14 @@ $$\begin{aligned} &= \int_{-\infty}^\infty \big( \hat{\mathcal{F}}\{f(x)\}\big)^* \: \hat{\mathcal{F}}\{g(x)\} \dd{k} \\ &= A^2 \int - \Big( \int f^*(x_1) \exp(- i s k x_1) \dd{x_1} \Big) - \Big( \int g(x) \exp(i s k x) \dd{x} \Big) + \Big( \int f^*(x') \: e^{- i s k x'} \dd{x'} \Big) + \Big( \int g(x) \: e^{i s k x} \dd{x} \Big) \dd{k} \\ - &= 2 \pi A^2 \iint f^*(x_1) g(x) \Big( \frac{1}{2 \pi} \int_{-\infty}^\infty \exp(i s k (x_1 - x)) \dd{k} \Big) \dd{x_1} \dd{x} + &= 2 \pi A^2 \iint f^*(x') \: g(x) \Big( \frac{1}{2 \pi} + \int_{-\infty}^\infty e^{i s k (x - x')} \dd{k} \Big) \dd{x'} \dd{x} \\ - &= 2 \pi A^2 \iint f^*(x_1) \: g(x) \: \delta(s (x_1 - x)) \dd{x_1} \dd{x} + &= 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} |