8.3 Schwartz space and its properties

Schwartz space is a crucial concept in harmonic analysis, consisting of smooth functions that decay rapidly at infinity. It's the perfect playground for Fourier transforms, allowing us to extend many important results to a broader class of functions.

The properties of Schwartz space, like its completeness and density in various function spaces, make it incredibly useful. It's the foundation for studying tempered distributions, which generalize functions and include important objects like the Dirac delta.

Definition and Properties of Schwartz Space

Rapidly Decreasing Functions and Schwartz Space

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• Schwartz space $\mathcal{S}(\mathbb{R}^n)$ consists of smooth functions $f \in C^{\infty}(\mathbb{R}^n)$ such that for all multi-indices $\alpha$ and $\beta$, the seminorm $\sup_{x \in \mathbb{R}^n} |x^{\alpha} D^{\beta} f(x)| < \infty$
• Functions in Schwartz space are rapidly decreasing, meaning they and all their derivatives decay faster than any polynomial at infinity
• Examples of functions in Schwartz space include $e^{-|x|^2}$, $\frac{1}{1+|x|^2}$, and $e^{-x^2}\sin(x)$
• Schwartz space is closed under multiplication by polynomials, differentiation, and Fourier transform

Seminorms and Topological Structure

• Seminorms $\rho_{\alpha,\beta}(f) = \sup_{x \in \mathbb{R}^n} |x^{\alpha} D^{\beta} f(x)|$ define the topology on Schwartz space
• The family of seminorms $\{\rho_{\alpha,\beta}\}_{\alpha,\beta}$ generates a locally convex topology on $\mathcal{S}(\mathbb{R}^n)$
• Convergence in Schwartz space means convergence with respect to all seminorms $\rho_{\alpha,\beta}$
• A sequence $\{f_k\}$ converges to $f$ in $\mathcal{S}(\mathbb{R}^n)$ if and only if $\rho_{\alpha,\beta}(f_k - f) \to 0$ for all $\alpha,\beta$
• Schwartz space is a Fréchet space, a complete metrizable locally convex topological vector space

Tempered Distributions and Density

Tempered Distributions

• Tempered distributions $\mathcal{S}'(\mathbb{R}^n)$ are continuous linear functionals on Schwartz space $\mathcal{S}(\mathbb{R}^n)$
• For $u \in \mathcal{S}'(\mathbb{R}^n)$ and $\varphi \in \mathcal{S}(\mathbb{R}^n)$, the action of $u$ on $\varphi$ is denoted by $\langle u, \varphi \rangle$
• Tempered distributions have a well-defined Fourier transform $\hat{u}$ given by $\langle \hat{u}, \varphi \rangle = \langle u, \hat{\varphi} \rangle$ for all $\varphi \in \mathcal{S}(\mathbb{R}^n)$
• Examples of tempered distributions include locally integrable functions with polynomial growth, Dirac delta function $\delta$, and derivatives of $\delta$

Density and Nuclear Space Properties

• Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$ for $1 \leq p < \infty$, meaning any $L^p$ function can be approximated by Schwartz functions
• $\mathcal{S}(\mathbb{R}^n)$ is dense in the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n)$ with the weak-* topology
• Schwartz space is a nuclear space, implying it has strong properties like the kernel theorem and the existence of a Gel'fand triple $\mathcal{S}(\mathbb{R}^n) \subset L^2(\mathbb{R}^n) \subset \mathcal{S}'(\mathbb{R}^n)$
• The nuclearity of Schwartz space allows for the extension of the Fourier transform to tempered distributions and the study of pseudodifferential operators