Examples
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 ƒ(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 in multiplied by the Fourier series for ƒ. This is the same as saying that differentiation is a multiplier operator with multiplier 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), where sgn is the signum function.
Finally another important example of a multiplier is the characteristic function of the unit ball in ℝn which arises in the study of "partial sums" for the Fourier transform (see Convergence of Fourier series).
Read more about this topic: Multiplier (Fourier Analysis)
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