Symbol class

Summary

A symbol class is a space of smooth functions $a(x, \xi)$ of a position $x$ and a frequency $\xi$. Its elements grow or decay like a fixed power of $\abs{\xi}$, and their derivatives in $\xi$ decay faster. The standard classes are the Hörmander classes $S^m_{\rho, \delta}$, where $m$ is the order. Symbols generalise the polynomials $\sum_\alpha a_\alpha(x) \xi^\alpha$ attached to differential operators. They are the building blocks of pseudo-differential operators and of Fourier integral operators.