I have learned all these things from H.Reiter’s book:Zbl 0165.15601 Reiter, H.
Classical harmonic analysis and locally compact groups. (English)
Oxford: At the Clarendon Press. XI, 200 p. (1968).

 

resp.

Zbl 0965.43001
Reiter, Hans; Stegeman, Jan D.
Classical harmonic analysis and locally compact groups. 2nd ed. (English)
[B] London Mathematical Society Monographs. New Series. 22. Oxford: Clarendon Press. xiii, 327 p. \sterling 60.00 (2000). ISBN 0-19-851189-2/hbk

 

1)       Proof it for the continuous functions with compact support, using uniform continuity!! (of course it is only valid for p < infty)

2)       multipliers are either “Fourier multipliers”, i.e. linear operators that can be written as multiplication operators on the Fourier transform side, engineers would say, that there is a transfer function (essentially what audio engineers are doing when they adjust to amplitude of certain frequency bands). Equivalentely, it is an operator commuting with translation (Translation invariant linear system) or again equiv. is a convolution matrix:

3)       Fourier algebra = linear space of all Fourier transforms of L1 = Lebesgue intergrable functions. These are all continuous and “small at infinity” (Riemann Lebesgue Lemma), but not all of those functions are in the Fourier algebra. In fact. it is not possible to characterize exactly the elements of the Fourier algebra (but there are good sufficient and necessary conditions).

 

HGFei  (see www.nuhag.eu for my group and work on time-frequency analysis)