How do I prove that the absolute value of CTFT of a positive continuous time signal is less than or equal to its value at central frequency(0)?


If $f(t)$ is a real non-negative function with Fourier transform

$$F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\tag{1}$$

the following must hold:

$$|F(\omega)|=\left|\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\right|\le\int_{-\infty}^{\infty}|f(t)|dt=\int_{-\infty}^{\infty}f(t)dt=F(0)$$

since $f(t)=|f(t)|$.

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