Given a list of variables $u_1,\dots,u_m\in\mathbb R$ and $v_1,\dots,v_n\in\mathbb R$ the standard convolution is defined

$$U*V(t)={\sum_{i}} u_iv_{t-i}.$$

Given a list of vectors $u_1,\dots,u_m\in\mathbb R^d$ and $v_1,\dots,v_n\in\mathbb R^d$ define vector-convolution $$U*V(t)={\sum_{i}}u_iv_{t-i}^T.$$

Is there a Fourier transform for this convolutions that converts this to a 'product' $\widetilde U(f)\cdot\widetilde V(f)$ from which we can perform inverse fourier transform to get the convolution?

Can we replace $u_iv_{t-i}^T$ by a polynomial or a rational function with $u_i,v_i\in\mathbb R^d$ arguments ($2d$ variables)?

Do these admit $FFT$`s?

`Fourier`

transforms are not invertible here. $\endgroup$ – T.... Aug 19 '19 at 20:57