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eliotsbowe
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Hello, I'm dealing with the proof of the theorem below:
x(t) \in L^1(\mathbb{R}), tx(t) \in L^1(\mathbb{R}) \Rightarrow \mathfrak{F}[x(t)] \in C^1(\mathbb{R})and\frac{\mathrm{d} }{\mathrm{d} \omega}\mathfrak{F}[x(t)](\omega) = \mathfrak{F}[-jtx(t)]I'm going to write down an interesting proof that I found, which is based on Lebesgue's dominated convergence theorem:
Let X(\omega) be the Fourier transform of x(t). Let's consider average rate of change of X:
\frac{X(\omega + \Delta \omega) - X ( \omega)}{\Delta \omega} = \int_{-\infty}^{+\infty}\frac{x(t) e^{-j(\omega + \Delta \omega)t}}{\Delta \omega}dt - \int_{-\infty}^{+\infty}\frac{x(t) e^{-j(\omega)t}}{\Delta \omega}dt = \int_{-\infty}^{+\infty}\frac{x(t) e^{-j(\omega + \Delta \omega)t}}{\Delta \omega} - \frac{x(t) e^{-j(\omega)t}}{\Delta \omega}dt =
=\int_{-\infty}^{+\infty} x(t) e^{-j \omega t} [ \frac{e^{-j \Delta \omega t} - 1} { \Delta \omega }] dt
Now there's a step I don't understand:
\forall \Delta \omega we have:
|x(t) e^{-j\omega t} \frac{e^{-j \Delta \omega t} - 1}{\Delta \omega}|=|tx(t) \frac{e^{-j \Delta \omega t} - 1}{-jt\Delta \omega }| = |tx(t)||\frac{e^{-j \Delta \omega t} - 1}{-jt\Delta \omega }| \leq |tx(t)|
(The thesis follows by noticing that tx(t) is a summable majorant for the integrandus, thus we can pass the \Delta \omega \to 0 limit inside the integral).
In particular, I don't understand the disequation:
|\frac{e^{-j \Delta \omega t} - 1}{-jt\Delta \omega }| \leq 1 , for any \Delta \omega
What am I missing?
Thanks in advance.
x(t) \in L^1(\mathbb{R}), tx(t) \in L^1(\mathbb{R}) \Rightarrow \mathfrak{F}[x(t)] \in C^1(\mathbb{R})and\frac{\mathrm{d} }{\mathrm{d} \omega}\mathfrak{F}[x(t)](\omega) = \mathfrak{F}[-jtx(t)]I'm going to write down an interesting proof that I found, which is based on Lebesgue's dominated convergence theorem:
Let X(\omega) be the Fourier transform of x(t). Let's consider average rate of change of X:
\frac{X(\omega + \Delta \omega) - X ( \omega)}{\Delta \omega} = \int_{-\infty}^{+\infty}\frac{x(t) e^{-j(\omega + \Delta \omega)t}}{\Delta \omega}dt - \int_{-\infty}^{+\infty}\frac{x(t) e^{-j(\omega)t}}{\Delta \omega}dt = \int_{-\infty}^{+\infty}\frac{x(t) e^{-j(\omega + \Delta \omega)t}}{\Delta \omega} - \frac{x(t) e^{-j(\omega)t}}{\Delta \omega}dt =
=\int_{-\infty}^{+\infty} x(t) e^{-j \omega t} [ \frac{e^{-j \Delta \omega t} - 1} { \Delta \omega }] dt
Now there's a step I don't understand:
\forall \Delta \omega we have:
|x(t) e^{-j\omega t} \frac{e^{-j \Delta \omega t} - 1}{\Delta \omega}|=|tx(t) \frac{e^{-j \Delta \omega t} - 1}{-jt\Delta \omega }| = |tx(t)||\frac{e^{-j \Delta \omega t} - 1}{-jt\Delta \omega }| \leq |tx(t)|
(The thesis follows by noticing that tx(t) is a summable majorant for the integrandus, thus we can pass the \Delta \omega \to 0 limit inside the integral).
In particular, I don't understand the disequation:
|\frac{e^{-j \Delta \omega t} - 1}{-jt\Delta \omega }| \leq 1 , for any \Delta \omega
What am I missing?
Thanks in advance.
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