Yesterday I finally figured out what they are, or rather, made what is the use of Fourier multipliers
. of explaining, we need two assumptions. temporal representation and 
spectral representation of a function are equivalent in that they contain the same amount of information. The second premise in functional analysis of linear operators often consider very simple, called
multiplier, the multiplier, call M, with a given function, say m, is defined as
Mf (x) = m (x ) f (x) These operators are designed for simplicity and because it is possible to illustrate many definitions and many theorems using these operators. For example, the specter of a multiplier (less than technical details) corresponds to the image of the function m. Now, since as we said temporal representation and spectral representation are equivalent, it is permissible to use multiplication operators for a function in frequency domain. That is, if \\ mu
is a function, the multiplier in the frequency domain is
\\ mathcal Mf (\\ omega) = \\ mu (\\ omega) f (\\ omega) Now we can define a Fourier multiplier
T, which is the temporal representation of a frequency multiplier space, ie Tf = \\ mathcal F ^ {-1} \\ mathcal M \\ mathcal F f said in Italian: we consider the spectral representation of a function f, the multiplied with a function
\\ mu
, and that we get the temporal representation. This series of operations defines a Fourier multiplier. Cui prodest? What is the benefit of these operators? But of course! What you can control exactly how these operators act on the spectrum! Some important examples : the Laplacian , the Hilbert transform , all
convolution operators, all
differential operators with constant coefficients are Fourier multipliers.
