# Generalized fourier transform and convolution?

Let $a(t)$ and $b(t)$ be two equal length sequences indexed by time index $t$.

We know that $a(t) * b(t)$ corresponds to $A(\omega) \odot B(\omega)$ in the frequency domain where $A(\omega)$ and $B(\omega)$ are row vectors indexed by frequency $\omega$ where $\odot$ is the Hadamard product of vectors $A(\omega)$ and $B(\omega)$ that are indexed by frequency $\omega$.

Generalizing this, supposing $a_{i}(t)$ and $b_{i}(t)$ are components of the time indexed vectors $a(t)$ and $b(t)$ then this still corresponds to $A(\omega) \odot B(\omega)$ where $\odot$ is the Hadamard product of matrices $A(\omega)$ and $B(\omega)$ that are indexed by frequency $\omega$.

Supposing we have $A(\omega)$ $\otimes$ $B(\omega)$ where $\otimes$ is the Kronecker product of matrices $A(\omega)$ and $B(\omega)$ that are indexed by frequency $\omega$, then does this correspond to any interesting convolution in the time domain?

Supposing we have $A(\omega)$ $\tilde\otimes$ $B(\omega)$ where $\tilde\otimes$ produces in the $i$th column the Kronecker product of column $i$ of matrices $A(\omega)$ and $B(\omega)$ that are indexed by frequency $\omega$(the matrix $A(\omega)$ $\tilde\otimes$ $B(\omega)$ is not square), then does this correspond to any interesting convolution in the time domain? I am actually interested in this more than the ordinary Kronecker Product.

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