Principal symbols and the Fourier Transform

Moreover, since the coefficients of equations (5) are constant at a given point $\mathbf{x}$, Renardy and Rogers (1992) say that the symbol can be interpreted in terms of the Fourier transform, with the Fourier transform of $u(\mathbf{x})$ being defined as

$$\hat u(\mathbf{\xi}):=\frac{1}{{(2\pi)}^{n/2}}\int_{\mathbb{R}^n} u(\mathbf{x})e^{-i\mathbf{\xi}\cdot\mathbf{x}} {\rm d}\mathbf{x} .$$

Then $L(i\mathbf{\xi})\hat u(\mathbf{\xi})$ is the Fourier transform of $L(D)u(\mathbf{x})$.

Since the concept of symbol and the Fourier transform are both important to FIO methods, this section provides an important bridge between the method of characteristics section and full application of FIO methods.