Does this gradient-adjusting variant of the FFT have a name?

Question

Consider the case of calculating an FFT of a not-quite-1-periodic function $$ t\to f(t) , t \in [0,1] , f(0) \neq f(1) $$ by transforming it into a periodic function :

$$f \to \hat f = f(t) - g(t) = f(t) - t\cdot (f(1)-f(0))$$

we can verify that $$\hat f(0) = f(0)\\\hat f(1) = f(1) - 1\cdot (f(1)-f(0)) = f(0)$$

This removes the step discontinuity at the edge and gives us (assuming f already continuous on the interior of [0,1]) a continuous 1-periodic function.

So consider the alternative forward-procedure

1. calculate and store $q = f(1)-f(0)$
2. calculate $\hat f(t) = f(t)-qt$
3. calculate and store $\hat F = \mathcal{F}(\hat f)$

And corresponding inverse procedure:

1. retrieve $\hat F$ and calculate $\hat f = \mathcal{F}^{-1}(\hat F)$
2. retrieve $q$ and calculate $f(t) = \hat f(t)+qt$

Does this procedure have any name?