Schwartz space ($\mathcal{S}({\mathbb{R}}^n)$)

The Schwartz space $\mathcal{S}({\mathbb{R}}^n)$ consists of smooth functions whose derivatives decay faster than every inverse polynomial. Its Fréchet topology is generated by the seminormsIt is dense in every $H^s({\mathbb{R}}^n)$.

These seminorms measure all polynomially weighted derivatives and generate the Fréchet topology of the Schwartz space. Finitely many such bounds control each weighted derivative's integrable norm by adding a spatial decay factor stronger than $⟨x{⟩}^{-n}$. Continuity of the Fourier transform follows by applying those bounds after differentiation and integration by parts. Equivalent families use powers of $⟨x⟩$ instead of individual monomials.

For a non-Archimedean local field $F$, the Schwartz-Bruhat space $\mathcal{S}(F)$ consists of the locally constant, compactly supported complex-valued functions on $F$.

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.

A Schwartz function is an element of the Schwartz space: it is a smooth function and every derivative decreases faster than every inverse polynomial. Equivalently, $x^{\alpha }{D}^{\beta }f(x)$ defines a bounded function for every pair of multi-indices $\alpha ,\beta$.

A tempered distribution is a continuous linear functional on the Schwartz space. Tempered distributions include functions of at most polynomial growth and admit a Fourier transform by duality.

For $T\in {\mathcal{S}}^{\prime }$ and $\psi \in \mathcal{S}$, define $(T\ast \psi )(x)=⟨T_y,\psi (x-y)⟩$. Translations are smooth in the Schwartz space, so every derivative is ${\partial }^{\alpha }(T\ast \psi )(x)=⟨T,{\partial }^{\alpha }\psi (x-\cdot )⟩$. The finite-seminorm continuity estimate for $T$ implies $|{\partial }^{\alpha }(T\ast \psi )(x)|\le C_{\alpha }(1+|x|{)}^m$ for one $m$. Thus the result is a smooth function defining a tempered distribution, but it need not be a Schwartz function: $1\ast \psi =1$ when $\int \psi =1$. If both factors are radial, testing the rotation action on the translated $\psi$ shows that the convolution is radial.

For $n>1$, a tempered distribution $T$ on ${\mathbb{R}}^n$ is radial when $T\circ R=T$ for every $R\in SO(n)$, where $⟨T\circ R,\phi ⟩=⟨T,\phi \circ R^t⟩$. This agrees with the usual radial function definition for regular smooth distributions, because the special orthogonal group acts transitively on spheres.

Choose a radial mollifier $\rho \in C_c^{\infty }$ with integral one and support in the unit ball, and a radial cutoff function $\chi \in C_c^{\infty }$ equal to one there. For a radial tempered distribution $T$, the functionsare radial Schwartz functions. Smoothness comes from convolution of a tempered distribution with a Schwartz function, and compact support comes from the cutoff. Pairing with $\phi$ gives $⟨f_j,\phi ⟩=⟨T,{\check{\rho }}_{1/j}\ast (\chi (\cdot /j)\phi )⟩$. With $q_m(\phi )={\max }_{|\beta |\le m}{\sup }_x(1+|x|{)}^m|{\partial }^{\beta }\phi (x)|$, the cutoff tail and the mean value theorem giveThe continuity estimate for $T$ therefore proves $f_j\to T$ even in the strong dual topology. Inserting a cutoff is essential because convolution alone need not give rapid decay.

A sequence $u_j\in {\mathcal{S}}^{\prime }$ converges weakly to $u$ if $⟨u_j,\phi ⟩\to ⟨u,\phi ⟩$ for every Schwartz function $\phi$. This is the weak-star topology of the continuous dual of the Schwartz space, and differs from choosing a strong dual topology defined by uniform convergence on bounded sets of test functions.

With $\hat{\phi }(\xi )=\int e^{-ix\cdot \xi }\phi (x)dx$, the Fourier transform of a tempered distribution is defined byContinuity of the Fourier transform on the Schwartz space makes this well defined. For regular integrable functions, Fubini's theorem shows that it agrees with the ordinary transform. The inverse convention has factor $(2\pi {)}^{-n}$ and the positive exponential.

A real simple pole is not locally integrable in the ordinary improper sense. Its symmetric Cauchy principal value defines a tempered distribution. For the $e^{-ikx}$ transform convention, contour indentation or sine integration gives the displayed nondecaying oscillatory contribution. Upper and lower boundary-value prescriptions instead change it by $\mp i\pi \delta (x-a)$ before transforming. A pole prescription must therefore be stated before claiming a Fourier asymptotic expansion.

For an orthogonal matrix $R$, a unit-Jacobian change of variables gives $\widehat{\phi \circ R^t}=\hat{\phi }\circ R^t$. The dual definition of the Fourier transform of a tempered distribution consequently givesSince the Fourier transform is invertible on the Schwartz space and its dual, a tempered distribution is invariant under a rotation group exactly when its transform is invariant. This includes radial tempered distributions.

For $D_j=-i{\partial }_j$, the Fourier transform of a tempered distribution satisfiesThe base cases follow from $⟨D_{j}u,\phi ⟩=i⟨u,{\partial }_j\phi ⟩$ and integration by parts in the Fourier integral. Iterating gives the multi-index formulas. The $D$ convention matters: for the unscaled derivative, $\widehat{{\partial }_{j}u}=i{\xi }_j\hat{u}$.

For $u(x)=\frac{1}{2}\log (1+x^2)$, the angular-frequency transform is the tempered distributionThe numerator cancels the singularity at zero and makes this integral absolutely convergent. Differentiating $u$ and using $\widehat{(1+x^2{)}^{-1}}=\pi e^{-|\xi |}$ determines this expression up to a Dirac delta distribution. That delta coefficient is zero: testing with the expanding Gaussian ${\phi }_n(x)=(2\sqrt{\pi }{)}^{-1}{e}^{-x^2/(4n)}$ makes both the proposed expression and $⟨u,{\hat{\phi }}_n⟩$ tend to zero, while ${\phi }_n(0)$ stays fixed. Changing the subtraction convention would change the delta coefficient.

A real distribution $u$ is positive when $u[\varphi ]\ge 0$ for every nonnegative test function $\varphi$. Every positive distribution has order zero and is represented locally by a positive measure.