Fourier Multipliers as Operators

Fourier analysis is especially useful for linear, translation-invariant operators. In physical space, such an operator may involve derivatives, integrals, or nonlocal interactions. In Fourier space, the same operation can often be described by multiplying each frequency by a scalar function.

The multiplier definition

Let \(f\) be a sufficiently regular function on \(\mathbb{R}^d\), and let \(a(\xi)\) be a prescribed function of the frequency variable. The Fourier multiplier operator \(T_a\) is defined by

\[ \widehat{T_a f}(\xi)=a(\xi)\widehat f(\xi). \]

Equivalently, if \(M_a\) denotes multiplication by \(a\) in frequency space, then

\[ T_a=\mathcal{F}^{-1}M_a\mathcal{F}. \]

This identity describes a three-step procedure:

1. take the Fourier transform of the input;
2. multiply every frequency component by \(a(\xi)\);
3. apply the inverse Fourier transform.

The function \(a\) is called the symbol of the operator. The symbol is not the Fourier transform of the operator treated as an ordinary function; it specifies how the operator acts after Fourier transformation.

Why Fourier modes simplify the operator

A pure Fourier mode has the form

\[ e^{i\xi\cdot x}. \]

For a multiplier operator, this mode is an eigenfunction:

\[ T_a e^{i\xi\cdot x}=a(\xi)e^{i\xi\cdot x}. \]

The operator therefore acts independently on each frequency. A superposition of modes is handled by applying the appropriate factor to each one, with no mixing between different frequencies.

Functions of differential operators

Fourier symbols also give a precise meaning to functions of differential operators. Under the Fourier transform, the positive operator \(-\Delta\) has symbol \(|\xi|^2\). Its square root is therefore defined by

\[ \widehat{(-\Delta)^{1/2}f}(\xi)=|\xi|\widehat f(\xi). \]

More generally, if \(q\) is a suitable scalar function, then \(q((-\Delta)^{1/2})\) is defined spectrally by the symbol \(q(|\xi|)\):

\[ \widehat{q((-\Delta)^{1/2})f}(\xi) =q(|\xi|)\widehat f(\xi). \]

This notation refers to functional calculus for an operator. It does not mean applying \(q\) pointwise to the value of a derivative.

Local and nonlocal behaviour

Polynomial symbols correspond to constant-coefficient differential operators. For instance, powers of \(|\xi|^2\) produce powers of \(-\Delta\). A non-polynomial symbol usually defines a nonlocal operator: the value of \(T_a f\) at one point can depend on values of \(f\) elsewhere.

When an inverse Fourier transform of \(a\) can be interpreted as a kernel \(K\), the multiplier may also be written formally as a convolution,

\[ T_a f=K*f. \]

This kernel representation can be useful, but it is conceptually different from the symbol. The symbol multiplies in frequency space, whereas the kernel convolves in physical space.

Invertibility and the zero frequency

If \(a(\xi)\) never vanishes and \(1/a(\xi)\) has suitable growth, an inverse can be defined through the reciprocal symbol:

\[ \widehat{T_a^{-1}g}(\xi)=\frac{1}{a(\xi)}\widehat g(\xi). \]

If the symbol vanishes at some frequencies, those modes lie in the kernel of the operator. In particular, when \(a(0)=0\), the constant mode must be handled separately. One may restrict to mean-zero functions, prescribe the constant mode, or use a generalized inverse on the remaining frequencies.

A practical reading rule

Whenever an operator is presented through a symbol, read the notation operationally:

\[ f\longmapsto \widehat f \longmapsto a\widehat f \longmapsto \mathcal{F}^{-1}(a\widehat f). \]

This viewpoint avoids several common confusions: an operator is not an ordinary function, nonlinear functions of operators are defined through spectral action, and multiplication in physical space should not be confused with multiplication in frequency space.