Frederic P Miller

Multiplier (Fourier analysis) Frederic P Miller

Publisher Marketing: Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 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. In the wider context, multiplier operators are special cases of spectral multiplier operators, which arise from the functional calculus of an operator.