Multiplier (Fourier analysis)

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 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). In the wider context, multiplier operators are special cases of spectral multiplier operators, which arise from the functional calculus of an operator (or family of commuting operators). They are also special cases of pseudo-differential operators, and more generally Fourier integral operators. There are natural questions in this field that are still open, such as characterizing the Lp bounded multiplier operators (see below). Multiplier operators are unrelated to Lagrange multipliers, except that they both involve the multiplication operation. For the necessary background on the Fourier transform, see that page. 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 f ( t ) . {displaystyle f(t).} After using integration by parts in the definition of the Fourier coefficient we have that So, formally, it follows that the Fourier series for the derivative is simply the Fourier series for f {displaystyle f} multiplied by a factor i n {displaystyle in} . This is the same as saying that differentiation is a multiplier operator with multiplier i n {displaystyle in} . An example of a multiplier operator acting on functions on the real line is the Hilbert transform. It can be shown that the Hilbert transform is a multiplier operator whose multiplier is given by the m ( ξ ) = − i sgn ⁡ ( ξ ) {displaystyle m(xi )=-ioperatorname {sgn} (xi )} , where sgn is the signum function. Finally another important example of a multiplier is the characteristic function of the unit cube in R n {displaystyle mathbb {R} ^{n}} which arises in the study of 'partial sums' for the Fourier transform (see Convergence of Fourier series).