Multiplier (Fourier Analysis) - Definition

Definition

Multiplier operators can be defined on any group G for which the Fourier transform is also defined (in particular, on any locally compact abelian group). The general definition is as follows. If is a sufficiently regular function, let denote its Fourier transform (where is the Pontryagin dual of G). Let denote another function, which we shall call the multiplier. Then the multiplier operator associated to this symbol m is defined via the formula

In other words, the Fourier transform of at a frequency is given by the Fourier transform of at that frequency, multiplied by the value of the multiplier at that frequency. This explains the terminology "multiplier".

Note that the above definition only defines implicitly; in order to recover explicitly one needs to invert the Fourier transform. This can be easily done if both f and m are sufficiently smooth and integrable. One of the major problems in the subject is to determine, for any specified multiplier m, whether the corresponding Fourier multiplier operator continues to be well-defined when f has very low regularity, for instance if it is only assumed to lie in an space. See the discussion on the "boundedness problem" below. As a bare minimum, one usually requires the multiplier m to be bounded and measurable; this is sufficient to establish boundedness on but is in general not strong enough to give boundedness on other spaces.

One can view the multiplier operator T as the composition of three operators, namely the Fourier transform, the operation of pointwise multiplication by m, and then the inverse Fourier transform. Equivalently, T is the conjugation of the pointwise multiplication operator by the Fourier transform. Thus one can think of multiplier operators as operators which are diagonalized by the Fourier transform.