What is the relation between a filter matrix and convolution Fourier basis?

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I read this post which is about why convolution is equal to multiplication in the frequency domain. But I can't get the second part of the answer. Is every transform equal to a new frequency domain? and is convolution equal to multiplication in every other domain? I can't get what is the relation between filter, eigenvector-eigenvalue and convolution in the mentioned answer. Could you please a little explain about it? Thanks