= Solution
The <Malgrange–Ehrenpreis theorem> states that \b[every nonzero constant-coefficient linear differential operator on <Euclidean space> has a distributional fundamental solution]. With $D_j=\partial_{x_j}$ and $P(D)=\sum_{|\alpha|\leq N}a_\alpha D^\alpha$, the conclusion is an $E\in\mathcal D'(\mathbb R^d)$ such that $P(D)E=\delta_0$. If $N=0$, simply take $E=a_0^{-1}\delta_0$; hence assume $N\geq1$.
We construct a <Hörmander staircase> in frequency space. The highest-degree homogeneous part $P_N$ is not identically zero on real vectors, since a <polynomial> vanishing on all real vectors has all coefficients zero. After an <orthogonal transformation> of coordinates, we can arrange that $P_N(e_1)\ne0$. Thus, for each real $\xi'\in\mathbb R^{d-1}$, the <polynomial>
$$
p(z,\xi')=P(iz,i\xi')
$$
has degree $N$ in $z$ and the same nonzero leading coefficient $b=i^NP_N(e_1)$, independent of $\xi'$. This constant <leading coefficient of a polynomial> is what makes a uniform staircase possible.
For a fixed $\xi'$, factor $p(z,\xi')=b\prod_{\nu=1}^N(z-z_\nu)$, counting repeated <roots of a polynomial>. Consider the $N+1$ heights $h_j=3j$, $j=0,\ldots,N$. A given root has imaginary part within distance less than one of at most one of these heights. By the <pigeonhole principle>, some height avoids every root, and then
$$
|p(s+ih_j,\xi')|=|b|\prod_\nu|s+ih_j-z_\nu|\geq|b|\qquad(s\in\mathbb R).
$$
This is <finite-height polynomial root avoidance>. Root labels need not be chosen continuously or even measurably. Instead define the closed sets
$$
B_j=\bigcap_{q\in\mathbb Q}\{\xi':|p(q+ih_j,\xi')|\geq|b|\},\qquad
A_j=B_j\setminus\bigcup_{\ell<j}B_\ell.
$$
Continuity in $s$ extends the inequality from rational $q$ to every real $s$. These <Borel sets> partition $\mathbb R^{d-1}$. The <Hörmander staircase> assigns the horizontal contour $s+ih_j$ over $A_j$; its height is bounded by $3N$, and the denominator has the uniform lower bound $|b|$. For $d=1$, the transverse space is a single point and only one horizontal contour is needed.
Use the <Fourier transform> convention $\widehat\varphi(\zeta)=\int e^{-ix\cdot\zeta}\varphi(x)\,dx$, with inverse factor $(2\pi)^{-d}$. Define the candidate <fundamental solution of a linear differential operator> by
$$
\langle E,\varphi\rangle=\frac1{(2\pi)^d}\sum_{j=0}^N\int_{A_j}\int_{\mathbb R}
\frac{\widehat\varphi(-s-ih_j,-\xi')}{P(i(s+ih_j),i\xi')}\,ds\,d\xi'.
$$
For a <test function> supported in a fixed <compact set> $K$, <Fourier decay in a bounded complex strip> gives, for any integer $M$,
$$
|\widehat\varphi(-\xi-ih_je_1)|\leq C_{K,M,N}(1+|\xi|^2)^{-M}
\max_{|\alpha|\leq2M}\|D^\alpha\varphi\|_\infty.
$$
Indeed, this is the real <Fourier transform> of $e^{-h_jx_1}\varphi(x)$ with reversed frequency, and repeated <integration by parts> with $(1-\Delta)^M$ proves the estimate. Taking $2M>d$ and using the denominator bound proves absolute convergence and a continuity estimate on $\mathcal D_K$. Thus $E$ is a <distribution>; no unsupported interpretation of a divergent inverse <Fourier transform> is being used.
For its <distributional derivatives>, the <formal transpose of a differential operator> is $P(-D)$. Since
$$
\widehat{P(-D)\varphi}(-\zeta)=P(i\zeta)\widehat\varphi(-\zeta),
$$
applying $P(D)$ cancels the denominator. For each fixed $\xi'$, the numerator is an <entire function> of $z$. The <Cauchy integral theorem>, applied to a rectangle between the lines $\operatorname{Im}z=0$ and $\operatorname{Im}z=h_j$, gives
$$
\int_{\mathbb R}\widehat\varphi(-s-ih_j,-\xi')\,ds
=\int_{\mathbb R}\widehat\varphi(-s,-\xi')\,ds.
$$
The two vertical edges tend to zero by the same bounded-strip decay. The resulting <contour deformation> is performed separately for each transverse frequency, so discontinuities of the staircase height introduce no additional boundary terms. Absolute convergence permits integration over the partition $A_j$. The <Fourier inversion theorem> then yields
$$
\boxed{\langle P(D)E,\varphi\rangle=\frac1{(2\pi)^d}\int_{\mathbb R^d}\widehat\varphi(-\xi)\,d\xi=\varphi(0).}
$$
This proves the <Malgrange–Ehrenpreis theorem>. Undoing the orthogonal change of coordinates gives the <fundamental solution of a linear differential operator> for the original operator; the <Dirac delta distribution> is unchanged by that change of coordinates.
For a one-dimensional operator $L=P(D)$ of degree $N\geq1$, with leading coefficient $a_N$, we may choose the single staircase contour below every pole. Take $\gamma>\max\{\operatorname{Re}\lambda:P(\lambda)=0\}$ and integrate on $\operatorname{Im}\zeta=-\gamma$. The corresponding formula is the <Bromwich contour> version of the construction above, after putting $\lambda=i\zeta$. Its poles all lie above the frequency contour. For $x<0$, close that contour downwards; the <exponential function> $e^{ix\zeta}$ decays and there are no enclosed poles. For $x>0$, close upwards, where the <exponential function> again decays, and apply the <residue theorem>. The bound $1/P(i\zeta)=O(|\zeta|^{-N})$ justifies the large arcs, including $N=1$ by the <Jordan lemma> away from their endpoints. Thus away from $x=0$ the <retarded fundamental solution> is $H(x)u(x)$, where
$$
\boxed{u(x)=\sum_{P(\lambda)=0}\operatorname*{Res}_{z=\lambda}\frac{e^{zx}}{P(z)}.}
$$
The sum is over distinct <roots of a polynomial>, with the residue including the whole multiplicity. For simple <characteristic roots of a constant-coefficient differential equation>, it reduces to $u(x)=\sum_\lambda e^{\lambda x}/P'(\lambda)$. Repeated roots give <exponential polynomial solutions of a constant-coefficient differential equation> through differentiation of $e^{zx}$.
To establish the equality also at the origin, rather than leave a possible point-supported term undecided, verify the <distributional jump formula for a Heaviside product>. The residue expression is an <entire function> of $x$, and
$$
P(D)u(x)=\sum_\lambda\operatorname*{Res}_{z=\lambda}e^{zx}=0.
$$
Its initial derivatives are
$$
u^{(j)}(0)=\sum_\lambda\operatorname*{Res}_{z=\lambda}\frac{z^j}{P(z)}
=0\quad(0\leq j\leq N-2),\qquad u^{(N-1)}(0)=a_N^{-1}.
$$
These identities follow by integrating over a large circle: for $j\leq N-2$ the integrand is $O(|z|^{-2})$, whereas for $j=N-1$ its coefficient of $z^{-1}$ is $a_N^{-1}$. The <distributional jump formula for a Heaviside product> therefore gives $L(Hu)=\delta_0$. Moreover, since $e^{-\gamma x}Hu$ is a <tempered distribution>, its <Fourier transform> satisfies
$$
P(i\xi+\gamma)\widehat{e^{-\gamma x}Hu}(\xi)=1.
$$
There are no real zeros of this polynomial, so its reciprocal is the transform. This is exactly the shifted-contour construction, proving equality there as a <distribution> as well. Hence
$$
\boxed{E=Hu,\qquad Lu=0,\qquad LE=\delta_0,\qquad\operatorname{supp}E\subset[0,\infty).}
$$
The sign of $a_N$ matters: for the operator in the preceding solution, $a_N=-1$ and the last initial derivative is $-1$.
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