0
$\begingroup$

I asked my professor to give a general definition of harmonic analysis and transform theory. He explained that given a Banach Algebra, say $L^1_m(\mathbb{R})$ with convolutions, transform theory arises when we consider elements of the dual of the Banach Algebra which are also homomophisms, in this case homomorphisms in $L^\infty_m(\mathbb{R})$. He said in the case of $L^1_m(\mathbb{R})$ with convolutions, if we look at such homomorphisms, we will find complex exponentials (cosines and sines), which are used to define the Fourier Transform. He explained that Transform theory extends beyond these Banach Algebras by several means (for example Plancherel's theorem to extend the Fourier transform to $L^2_m(\mathbb{R})$).

Now, I am not sure I understood him correctly, but I would like to find references that show these results. To start, I am looking for a proof of the result for the $L^1_m(\mathbb{R})$ case.

Any suggestions are welcome.

$\endgroup$
0
$\begingroup$

This material turned out to be all included in chapter 1 of Rudin's Fourier Analysis on Groups.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.