Frequency Support of Window Functions
A window function $w(t)$ satisfies $w(t)\ge 0$. The value of a window's frequency response $W(f)$ at DC ($f=0$) equals its integral
$$W(0)=\int_{-\infty}^{\infty}w(t)dt&gt;0\tag{1}$$
which is clearly greater than zero because $w(t)\ge 0$. For all other frequencies we obtain the following bound:
$$|W(f)|=\left|\int_{-\infty}^{\infty}w(t)e^{-j2\pi ft}dt\right|\le\int_{-\infty}^{\infty}\left|w(t)e^{-j2\pi ft}\right|dt=\int_{-\infty}^{\infty}w(t)dt=W(0)\tag{2}$$
Eq. $(2)$ means that $|W(f)|$ cannot be greater than its DC value. Actually, it can be shown that $(2)$ is a strict inequality for any $f\neq 0$: $|W(f)|&lt;W(0)$ for any $f&gt;0$. This shows that $W(f)$ has a lowpass characteristic.
To give a more intuitive explanation, a rectangular window's frequency response is a sinc function, which has a lowpass characteristic. All other windows are just smoother versions of a rectangular window, which means that their spectra decay faster than a sinc. I.e., non-rectangular windows have an even more pronounced low pass characteristic than a rectangular window.