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  "body": "\n## Summary\n\nA symbol class is a space of smooth functions $a(x, \\xi)$ of a position $x$ and a frequency $\\xi$.\nIts elements grow or decay like a fixed power of $\\abs{\\xi}$, and their derivatives in $\\xi$ decay faster.\nThe standard classes are the Hörmander classes $S^m_{\\rho, \\delta}$, where $m$ is the order.\nSymbols generalise the polynomials $\\sum_\\alpha a_\\alpha(x) \\xi^\\alpha$ attached to differential operators.\nThey are the building blocks of pseudo-differential operators and of Fourier integral operators.\n\n## Explanations\n\n### L1 human\n\nA differential operator such as $\\partial_x^2 + 1$ acts on a wave $e^{ix\\xi}$ by multiplying it by the polynomial $1 - \\xi^2$, its symbol.\nAllowing more general functions of $\\xi$ in place of polynomials leads to more general operators, for example a square root of $1 - \\Delta$.\nTo keep a good calculus one asks that the symbol behaves like a power $\\abs{\\xi}^m$ for large $\\xi$, and that each derivative in $\\xi$ lowers this power by one.\nFor instance $(1 + \\abs{\\xi}^2)^{s/2}$ is a symbol of order $s$.\n",
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