In engineering a common practice is to "window" a signal (by multiplying a function which decays smoothly at each end) before applying a Fourier transform. Windowing is done to avoid false frequency contributions at the boundary of the signal, which usually is discontinuous.
So recently when I asked a question about Reducing the Gibbs phenomenon when using Fourier transforms, I learned about kernels to apply, for instance the Fejér kernel - which make the reconstructed signal smoother at any number of frequency components. Then it suddenly struck me that multiplying with a kernel in the Fourier domain is practically the same as deciding portion of a signal and windowing in the spatial domain prior to Fourier transformation. The Dirichlet kernel being windowing with the "box" function, Fejér with a linearly decaying "triangle" and so on.
Applying the Dirichlet kernel for frequencies up to $\omega_0$: multiply all frequencies $|\omega| < \omega_0$ by 1, all $|\omega| \geq \omega_0$ by 0, followed by iFFT. I.e. "windowing" with a box function in the frequency domain before iFFT. Multiplying with boxes in the fourier domain corresponds to convolution with sincs in spatial domain - in other words this is the standard "reconstruction" algorithm for FFT in signal processing.
Applying the Fejér kernel for frequencies up to $\omega_0$: this is the arithmetic mean value of all the Dirac kernels up to $\omega_0$, adding boxes of same height but increasing base builds a triangle which we pointwise multiply in the Fourier domain followed by iFFT. Since the triangle function in the Fourier domain is the convolution of two boxes it corresponds to convolution with the product of two sincs in the spatial domain - the power of two which creates "positivity", i.e. contribution of each term in this reconstruction algorithm is ensured to be positive.
So when applying the kernel we can choose to convolve with functions from the sinc family in the spatial domain or multiply with functions in the frequency domain. If we choose to multiply in the fourier domain, this becomes similar to what a "windowing" of the function would do to the spatial domain. The kernel having limited support and different decline towards 0 giving different properties in the "other domain".
Is there any intuitive way to explain this connection between windowing in the standard domain and choosing a reconstruction kernel by windowing in the Fourier domain?