Regular polynomial field functional algebra

For a continuous compactly supported $u:{\mathbb{R}}^m\to \mathbb{C}$, write $u=u_{\mathrm{R}}+i{u}_{\mathrm{I}}$ for its real-coordinate decomposition and define its complex Riemann integral by $\int u:=\int u_{\mathrm{R}}+i\int u_{\mathrm{I}},$ where the real integrals are over a closed rectangle containing the support. Equivalently, use any sufficiently large cube $[-R,R{]}^m$; the value stabilizes as $R$ increases because $u$ vanishes outside its compact support. Each real component integral exists by Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable. The complex modulus is $\mathrm{\mid }a+ib\mathrm{\mid }=(a^2+b^2{)}^{1\mathrm{/}2}$; the triangle inequality for Riemann sums gives $\mathrm{\mid }\int u\mathrm{\mid }\le \int \mathrm{\mid }u\mathrm{\mid }$. Complex linearity and product-rectangle Fubini follow by applying the real integral rules to the real and imaginary components; in particular, use Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections componentwise. Differentiation under a compact parameter integral likewise means applying Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral to both real components. These conventions apply to every integral below.

Let $d\ge 1$, $X={\mathbb{R}}^d$, and $n\ge 1$. Write $\mathcal{D}(X^n)=C_c^{\mathrm{\infty }}(X^n;\mathbb{C})$ for the complex test-function space of smooth compactly supported functions (Test function space d of an open set), with its LF topology (Test function topology). The convergence criterion used below is that convergent test sequences have one common compact support and converge uniformly with every derivative (Sequential convergence in test function space). Here $C^{\mathrm{\infty }}$ means every coordinate derivative exists and is continuous ($C^k$ maps and multi-index derivative notation in Euclidean space). All integrals are compact Riemann integrals: each integrand is continuous with compact support, so restrict it to a compact rectangle and apply Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable. Let ${\mathcal{D}}_{\mathrm{s}\mathrm{y}\mathrm{m}}(X^n)$ be the subspace of kernels invariant under every permutation of their $n$ $X$-variables. Fields are real smooth configurations $\varphi \in C^{\mathrm{\infty }}(X;\mathbb{R})$. For an integer $N\ge 0$ and a finite family $f_0\in \mathbb{C}$, $f_n\in {\mathcal{D}}_{\mathrm{s}\mathrm{y}\mathrm{m}}(X^n)$ for $1\le n\le N$, define $F_f(\varphi )=f_0+{\sum }_{n=1}^N{\int }_{X^n}{f}_n(x_1,\dots ,x_n){\prod }_{j=1}^n\varphi (x_j)d{x}_1\cdots d{x}_n\mathrm{.}$ The pairing is complex-bilinear, with no conjugation on the kernel or field. An element of ${\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{g}}(X)$ is this finite symmetric coefficient family, evaluated on fields by $F_f$. Pointwise sums, scalar multiples, and products again have finite symmetric compactly supported coefficient families: for products, expand the two finite sums and symmetrize each tensor product $f_m\otimes g_n$. Thus these elements form an algebra; the product-integral identity follows from Riemann Fubini on a rectangle containing the compact supports (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

For $k=0$, set $F_f^{(0)}(\varphi )=F_f(\varphi )$. For $1\le k\le N$, the $k$-th functional derivative kernel assigned to this coefficient family is $F_f^{(k)}(\varphi )(x_1,\dots ,x_k)={\sum }_{n=k}^N\frac{n!}{(n-k)!}{\int }_{X^{n-k}}{f}_n(x_1,\dots ,x_n){\prod }_{j=k+1}^n\varphi (x_j)d{x}_{k+1}\cdots d{x}_n\mathrm{.}$ For $k>N$ it is zero. Each displayed kernel is smooth, symmetric and compactly supported in its $k$ remaining variables: compactness follows by projection of the finite union of coefficient-kernel supports, and derivatives pass through the finite-compact integrals by Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral. In the term $n=k$, the empty integral and empty product mean simply $f_k(x_1,\dots ,x_k)$.

For real smooth directions $h_1,\dots ,h_k$, the pairing of $F_f^{(k)}(\varphi )$ with $h_1\otimes \cdots \otimes h_k$ is the coefficient of $t_1\cdots t_k$ in $F_f(\varphi +{\sum }_{j=1}^k{t}_j{h}_j)$. For each degree-$n$ term, expanding the finite product assigns the $k$ distinct directions to the $n$ factors in $n!\mathrm{/}(n-k)!$ ways; symmetry of $f_n$ makes the corresponding integrals equal. This verifies the displayed kernel's functional-derivative meaning by a finite polynomial calculation.

The scope is exactly smooth compact coefficient kernels. A local functional whose higher derivative kernels are distributions on a thin diagonal is not included merely because its field expression is polynomial.