Smooth torus test functions have rapidly convergent Fourier pairings

Put $\tilde{\phi }(t)=\phi (At)$, which is smooth and ${\mathbb{Z}}^3$-periodic. Change variables to the unit cube and integrate successively in the three coordinates using [F8]. In $t$ coordinates the physical Laplacian is the constant-coefficient operator ${\mathrm{\Delta }}_t={\sum }_{r,s}(A^{-1}{A}^{-T}{)}_{rs}{\mathrm{\partial }}_{t_r}{\mathrm{\partial }}_{t_s}$. Repeatedly apply [F3] to the real and imaginary parts in each coordinate; periodicity makes every opposite-face boundary term cancel, and the character is an eigenfunction with eigenvalue $-\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2$. Thus for every integer $m\ge 1$, $\widehat{\phi }(k)=(1+\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^{-m}\widehat{(1-\mathrm{\Delta }{)}^m\phi }(k),\mathrm{\mid }\widehat{\phi }(k)\mathrm{\mid }\le (1+\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^{-m}{\sup }_{x\in F}\mathrm{\mid }(1-\mathrm{\Delta }{)}^m\phi (x)\mathrm{\mid }\mathrm{.}$ The last inequality follows because the Fourier coefficient is the normalized integral over the cell [F1].

Since $\mathrm{\mid }{\kappa }_k\mathrm{\mid }\ge 2\pi \mathrm{\mid }k\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{A}^T{\mathrm{\parallel }}_2$, the number of integer vectors with $j\le \mathrm{\mid }{\kappa }_k\mathrm{\mid }<j+1$ is at most $(2\mathrm{\parallel }{A}^T{\mathrm{\parallel }}_2(j+1)\mathrm{/}(2\pi )+3{)}^3$. For any fixed $q$, choose an integer $m>(q+4)\mathrm{/}2$; the shell sum of $\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^q(1+\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^{-m}$ is then bounded by a constant times a convergent p-series [F4]. Hence ${\sum }_k\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^q\mathrm{\mid }\widehat{\phi }(k)\mathrm{\mid }<\mathrm{\infty }$.

Enumerate ${\mathbb{Z}}^3$ by increasing ${\max }_r\mathrm{\mid }{k}_r\mathrm{\mid }$ and lexicographically within each finite shell. For each derivative order, step 2.1 and [F5] make the Fourier series and its formal derivative series uniformly convergent. Apply [F6] to the real and imaginary parts along each coordinate line; the uniform derivative limits are the derivatives of the sum. By [F2], termwise integration shows that the sum has the same Fourier coefficients as $\phi$; [F7] makes the continuous difference zero. Each derivative series likewise has the coefficients of the corresponding derivative of $\phi$, by periodic integration by parts [F3], so [F7] identifies it with that derivative. Hence the Fourier partial sums converge in every smooth seminorm.

Choose $m>(s+4)\mathrm{/}2$ in step 1.1. The shell count of step 2.1 gives ${\sum }_k\mathrm{\mid }{a}_k\widehat{\phi }(-k)\mathrm{\mid }\le C{\sup }_F\mathrm{\mid }(1-\mathrm{\Delta }{)}^m\phi \mathrm{\mid }{\sum }_k(1+\mathrm{\mid }{\kappa }_k\mathrm{\mid }{)}^s(1+\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^{-m}<\mathrm{\infty }\mathrm{.}$ This is a bound by the seminorm ${\sup }_F\mathrm{\mid }(1-\mathrm{\Delta }{)}^m\phi \mathrm{\mid }$, a finite sum of the smooth seminorms in the source-data definition, so $T_a$ is a continuous linear functional. If a periodic distribution has the same Fourier coefficients as $T_a$, their difference vanishes on every character and hence on every trigonometric polynomial; by the smooth convergence of step 3.1 and continuity of the distribution, it vanishes on every smooth test function. Therefore the coefficients determine it.