Multiplier (Fourier Analysis) - Multiplier Operators On Common Groups

Multiplier Operators On Common Groups

We now specialize the above general definition to specific groups G. First consider the unit circle ; functions on G can thus be thought of as -periodic functions on the real line. In this group, the Pontryagin dual is the group of integers, . 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 of numbers, and the operator associated to this multiplier is then given by the formula

at least for sufficiently well-behaved choices of the multiplier and the function f.

Now let G be a Euclidean space . Here the dual group is also Euclidean, and the Fourier and inverse Fourier transforms are given by the formulae

A multiplier in this setting is a function, and the associated multiplier operator is defined by

again assuming sufficiently strong regularity and boundedness assumptions on the multiplier and function.

In the sense of distributions, there is no difference between multiplier operators and convolution operators; every multiplier T can also be expressed in the form for some distribution K, known as the convolution kernel of T. In this view, translation by an amount x₀ is convolution with a Dirac delta function δ(· − x₀), differentiation is convolution with δ'. Further examples are given in the table below.