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.
Read more about this topic: Multiplier (Fourier Analysis)
Famous quotes containing the word definition:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)