The Schwartz–Bruhat function, often simply referred to as the Schwartz function, is a type of smooth function that is rapidly decreasing. Specifically, it belongs to the space of smooth functions that decay faster than any polynomial as one approaches infinity. This type of function is especially important in various areas of analysis, particularly in the fields of distribution theory, Fourier analysis, and partial differential equations.
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A Schwartz-Bruhat function on a non-Archimedean local field is a locally constant, compactly supported complex-valued function, or an element of the Schwartz-Bruhat space. A sufficiently small open additive subgroup leaves it invariant under translation, and only finitely many cosets meet its support.