Schwartz function

速降函数

Analysis, Measure & PDE · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A Schwartz function on R^n is a smooth complex-valued function phi such that sup_x |x^alpha partial^beta phi(x)| is finite for every pair of multi-indices alpha,beta.

速降函数是欧氏空间上的光滑函数，任意阶偏导数乘任意坐标多项式后都保持有界。

Example

$\varphi(x)=e^{-2x^2}$ is Schwartz, since each derivative is a polynomial times that Gaussian.

Usage and distinctions

Rapid decay is required for every derivative; ordinary integrability or decay of the function alone is insufficient.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Mikko Salo — Fourier Analysis and Distribution Theory§3.1, definition and equation (3.1) · Checked 2026-10-14

Translation sources

1. 电子科技大学 — 泛函分析教学大纲第三章，了解5：速降函数和缓增分布 · Checked 2026-10-14

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