In
Fourier analysis, a
multiplier operator is a type of
linear operator, or transformation of
functions. These operators act on a function by altering its
Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the
multiplier or
symbol. Occasionally, the term "multiplier operator" itself is shortened simply to "multiplier". In simple terms, the multiplier reshapes the frequencies involved in any function. This class of operators turns out to be broad: general theory shows that a translation-invariant operator on a
group which obeys some (very mild) regularity conditions can be expressed as a multiplier operator, and conversely. Many familiar operators, such as
translations and
differentiation, are multiplier operators, although there are many more complicated examples such as the
Hilbert transform. In
signal processing, a multiplier operator is called a "
filter", and the multiplier is the filter's
frequency response (or
transfer function).