These operators act on a function by altering its Fourier transform.
[1] In simple terms, the multiplier reshapes the frequencies involved in any function.
In signal processing, a multiplier operator is called a "filter", and the multiplier is the filter's frequency response (or transfer function).
There are natural questions in this field that are still open, such as characterizing the Lp bounded multiplier operators (see below).
Additional important background may be found on the pages operator norm and Lp space.
In the setting of periodic functions defined on the unit circle, the Fourier transform of a function is simply the sequence of its Fourier coefficients.
To see that differentiation can be realized as multiplier, consider the Fourier series for the derivative of a periodic function
An example of a multiplier operator acting on functions on the real line is the Hilbert transform.
Finally another important example of a multiplier is the characteristic function of the unit cube in
Note that the above definition only defines Tf implicitly; in order to recover Tf explicitly one needs to invert the Fourier transform.
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 Lp space.
As a bare minimum, one usually requires the multiplier m to be bounded and measurable; this is sufficient to establish boundedness on
Equivalently, T is the conjugation of the pointwise multiplication operator by the Fourier transform.
We now specialize the above general definition to specific groups G. First consider the unit circle
The Fourier transform (for sufficiently regular functions f) is given by and the inverse Fourier transform is given by A multiplier in this setting is simply a sequence
is defined by again assuming sufficiently strong regularity and boundedness assumptions on the multiplier and function.
The following table shows some common examples of multiplier operators on the unit circle
The following table shows some common examples of multiplier operators on Euclidean space
This problem is considered to be extremely difficult in general, but many special cases can be treated.
The problem depends greatly on p, although there is a duality relationship: if
The Riesz-Thorin theorem shows that if a multiplier operator is bounded on two different Lp spaces, then it is also bounded on all intermediate spaces.
Parseval's theorem allows to solve this problem completely and obtain that a function m is an L2(G) multiplier if and only if it is bounded and measurable.
In this general case, necessary and sufficient conditions for boundedness have not been established, even for Euclidean space or the unit circle.
Results that give sufficient conditions for boundedness are known as multiplier theorems.
be a bounded function that is continuously differentiable on every set of the form
The proofs of these two theorems are fairly tricky, involving techniques from Calderón–Zygmund theory and the Marcinkiewicz interpolation theorem: for the original proof, see Mikhlin (1956) or Mikhlin (1965, pp. 225–240).
[3] They also provided a necessary and sufficient condition which is valid without the compact support assumption on
However, both the Marcinkiewicz and Mikhlin multiplier theorems show that the Hilbert transform is bounded in Lp for all 1 < p < ∞.
From the Marcinkiewicz multiplier theorem (adapted to the context of the unit circle) we see that any such sequence (also assumed to be bounded, of course)[clarification needed] is a multiplier for every 1 < p < ∞.
However, in 1972, Charles Fefferman showed the surprising result that in two and higher dimensions the disk multiplier operator