Mathematical expression of a sliding window function

Question

I was trying to prove an equivalence of a function with another function which is performed in a sliding window process.

For example, the short term fourier transform (STFT) of an $N$-length signal can be done by taking a sliding window function of Fourier transform (FT) of arbitrary length $w$ along time $t$.

I just wanted to know how this sliding window process is expressed mathematically. If there are other simpler example, I would love to learn them.

Answer 1

It seems that convolution answers this.

$[f*g]\equiv\int_{0}^{t}f(\tau)g(t-\tau)d\tau$

where two funcions $f$ and $g$ convolved over a finite range $[0,t]$.

$g$ acts like a moving window over $f$.