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Multiplication by a test function is bounded on every real-index Sobolev space. Its Fourier transform acts by convolution with a Schwartz function; frequency-weight estimates and Young's convolution inequality give the norm bound.

Consider $u_{tt}=c^2{u}_{xx}-c^2{A}^{2}u$ with $A,c>0$, initial displacement $f$ a Schwartz function, and zero initial velocity. With $a(k)={\int }_{\mathbb{R}}f(x)e^{-ikx}dx$,For a fixed $|V|<c$, set $s=\sqrt{c^2-V^2}$ and $k_0=AV/s$. The phases $Vk\pm \omega (k)$ have their unique stationary points at $\mp k_0$, with stationary values $\pm As$ and curvature magnitudes $d_0=s^3/(A{c}^2)$. The one-dimensional stationary-phase formula givesFor a real even initial profile, $a$ is real and even, and the bracket reduces to $2a(k_0)\cos (Ast+\pi /4)$. Thus the ray oscillation has angular frequency$As$ and generic decay$t^{-1/2}$. If the sampled amplitudes vanish, the displayed additive error still makes sense. The estimate is for a fixed subluminal ray; it is not uniform as $|V|\uparrow c$, where the selected wavenumbers escape to infinity.

For every $ϵ>0$, choose a nonzero smooth spectral bump supported in $(-{ϵ}_0,{ϵ}_0)$ with ${ϵ}_0<\min (ϵ,\pi )$, and let $h$ be its inverse Fourier transform. Then $g$ is a nonzero Schwartz function, vanishes on all integer samples, and its transform is the difference of bumps shifted to $\pi$ and $-\pi$. Its support lies strictly within $[-\pi -ϵ,\pi +ϵ]$. Thus even a small bandwidth increase destroys uniqueness from unit-spaced samples.

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.