The dimension formulas attain the Cauchy data

Differentiated finite expansion. Suppose $n=2k+1$ and put $h_f(x,t)=M_f(x,ct)$, using the signed-radius extension of [F2]. This is $C^{k+1}$ and even when $f\in C^{k+1}$, so $h_f(x,0)=f(x)$ and ${\mathrm{\partial }}_t{h}_f(x,0)=0$. By [F1], $T_f(x,t):=D_t^{k-1}(t^{2k-1}{h}_f(x,t))={\sum }_{j=0}^{k-1}{\alpha }_{k,j}{t}^{j+1}{\mathrm{\partial }}_t^j{h}_f(x,t)$, where $a:={\alpha }_{k,0}=(2k-1)!!$. Differentiate this finite sum itself: $T_f^{\mathrm{\prime }}={\sum }_j{\alpha }_{k,j}((j+1)t^j{h}_f^{(j)}+t^{j+1}{h}_f^{(j+1)})$ and $T_f^{\mathrm{\prime }\mathrm{\prime }}={\sum }_j{\alpha }_{k,j}(j(j+1)t^{j-1}{h}_f^{(j)}+2(j+1)t^j{h}_f^{(j+1)}+t^{j+1}{h}_f^{(j+2)})$, with the first summand omitted for $j=0$. Every derivative used has order at most $k+1$. Continuity and $h_f^{\mathrm{\prime }}(x,0)=0$ give $T_f\to 0$, $T_f^{\mathrm{\prime }}\to af(x)$ and $T_f^{\mathrm{\prime }\mathrm{\prime }}\to 0$, uniformly for $x$ in compact sets. For $T_f^{\mathrm{\prime }\mathrm{\prime }}$, the only terms without a positive power of $t$ are constant multiples of $h_f^{\mathrm{\prime }}$, which vanish at zero. These formulas never differentiate an unspecified error term.

Odd-dimensional data. The odd formula is $u=a^{-1}(T_{u_0}^{\mathrm{\prime }}+T_{u_1})$ and $u_t=a^{-1}(T_{u_0}^{\mathrm{\prime }\mathrm{\prime }}+T_{u_1}^{\mathrm{\prime }})$. Step 1.1 applies to both data, since their classes are at least $C^{k+1}$. It follows that $u\to u_0$ and $u_t\to u_1$, locally uniformly in $x$. The even smooth signed-radius means in step 1.1 also show that $T_f$ and its first two derivatives extend continuously through zero.

Even-dimensional data by descent. For $n=2k$, extend the data cylindrically to ${\mathbb{R}}^{n+1}$. Their differentiability classes are exactly those of the odd formula in dimension $n+1=2k+1$. The construction in The even-dimensional wave formula by descent identifies the even solution with the restriction of that odd solution to the last coordinate zero. The limits of step 2.1 therefore apply without differentiating a singular ball weight.

The locally uniform displacement limit and continuity of $u_0$ give joint continuity of the extension $u(x,0)=u_0(x)$. The velocity limit holds as stated. The cases $n=3$ and $n=2$ are Kirchhoff and Poisson by [F3].