A linear Whitney–Hadamard operator, with smooth dependence on parameters

Let $P$ be a real normed space (a normed additive commutative group with a real normed space structure). The assertion is the existence of a single operator $W$ on complex-valued functions of one real variable, $W : (\mathbb{R} \to \mathbb{C}) \to (\mathbb{R} \to \mathbb{C})$, with three properties. First, $W$ is linear on smooth compactly supported functions: for all $f, g : \mathbb{R} \to \mathbb{C}$ that are $C^\infty$ with compact support and all $a, b \in \mathbb{C}$, one has $W(x \mapsto a f(x) + b g(x)) = (x \mapsto a\,(Wf)(x) + b\,(Wg)(x))$ as functions. Secondly, for every $C^\infty$ compactly supported $f$, the function $Wf$ is again $C^\infty$ with compact support, and: if $f$ is even, i.e. $f(-x) = f(x)$ for all real $x$, then $(Wf)(x^2) = f(x)$ for all real $x$; if $f$ is odd, i.e. $f(-x) = -f(x)$ for all real $x$, then $x \cdot (Wf)(x^2) = f(x)$ for all real $x$, the factor $x$ being its image in $\mathbb{C}$. Thirdly, $W$ acts well on families: for every $C^\infty$ compactly supported $H : \mathbb{R} \times P \to \mathbb{C}$, the function $(y, p) \mapsto \bigl(W(x \mapsto H(x, p))\bigr)(y)$ on $\mathbb{R} \times P$ is $C^\infty$ with compact support.

This packages Whitney's theorem on even smooth functions and the odd-function (Hadamard) variant into one operator that is simultaneously linear on test functions and compatible with smooth compactly supported families over a parameter space $P$. It is used in the analysis of archimedean weight characters for $\mathrm{GL}_2(\mathbb{R})$, in AutomorphicForm.GL2Real.exists_linear_entrySlice_archWeightChar_one_splitTransform_eq and AutomorphicForm.GL2Real.exists_linear_entrySlice_archWeightChar_zero_splitTransform_eq, where even and odd slices of a test function must be rewritten as functions of $x^2$ with control of smoothness and supports in the remaining variables.