Let $S_+ = \mathbb{R} \times \{t>0\}$ and $\,f \in C^{\infty}(S^+)$ such that $\,f$ and all its derivatives have continuous limits as $t \to 0^+$.
In this paper the author defines the following extension of $f$ for $t<0$: \begin{equation} Ef(x,t) = \sum_{k=1}^{\infty} a_k \,\phi(b_k t)\, f(x,b_k t) \end{equation} where
$\phi \in C^{\infty}(\mathbb{R})$ with: $\phi(t)=1$ for $0\leq t\leq1$ and $\,\phi(t)=0$ for $t\geq$2;
${a_k}, {b_k}$ such that:
- $b_k < 0$ and $b_k \to - \infty$,
- $\sum_{k=1}^\infty|a_k||b_k|^n <\infty$ for $\, n = 0,1, 2, \dots$,
- $\sum_{k=1}^\infty (a_k)(b_k)^n = 1$ for $\, n = 0,1, 2, \dots$.
The existence of sequences $\{a_k\}, \{b_k\}$ such that 2. holds is proved in the paper.
My observations
- Since $b_k \to -\infty$, for each $t<0$ the sum in $Ef$ is finite and therefore $Ef$ is pointwise well defined.
- If $C$ is a compact set in $\mathbb{R}\times(-\infty,0)$ and $\bar{t} = \max_{(x,t) \in C} t$, let $\bar{N}$ be such that for every $k \geq \bar{N}$ one has $b_k \bar{t} > 2$. Then since $b_k t \geq b_k \bar{t} > 2 $ for all $t \in C$, it follows that: $$ Ef(x,t) = \sum_{k=1}^\bar{N} a_k \,\phi(b_k t)\, f(x,b_k t)\qquad \forall (x,t) \in C $$ It follows that on every compact set $C \subset \mathbb{R}\times(-\infty,0)$ $Ef$ is uniform limit of functions in $C^\infty(\mathbb{R}\times(-\infty,0))$. Then it easy to see that the restriction of $Ef$ to $t<0$ is in $C^{\infty}(\mathbb{R}\times(-\infty,0))$.
My question:
I have shown that $Ef(t,x)$ is well defined for every $(t,x) \in R^2$, moreover the restriction of $Ef$ to $t<0$ is in $C^{\infty}(\mathbb{R}\times(-\infty,0))$. I'm left with $Ef(t,x)$ has derivative as $t\to 0^-$.
Question: How to prove that $\lim_{t\to 0^-} Ef(x,t)$ and $ \lim_{t\to 0^-} \partial_t^n \partial_x^j Ef(x,t)$ exist finite?
Remarks
a) I believe that to answer the question "dominated convergence" using the second point of 2.2 is what is needed, but I'm missing the details.
b) If the limit exist then they agree to those for $t\to 0^+$, since using the third point in 2.: $$ \lim_{t\to 0^-} Ef(x,t) = \sum_{k=1}^{\infty} \lim_{t\to 0^-} a_k \,\phi(b_k t)\, f(x,b_k t) = f(x, 0^+) \sum_{k=1}^{\infty} a_k = f(x, 0^+)$$ $$ \lim_{t\to 0^-} \partial_t^n \partial_x^j Ef(x,t) = \sum_{k=1}^{\infty} \lim_{t\to 0^-} a_k \sum_{j \leq n} C_{j,k} \partial_t^{n-j} \,\phi(b_k t)\,\partial_t^j \partial_x^j f(x,b_k t) = \partial_t^m \partial_x^n f(x, 0^+) \sum_{k=1}^{\infty} a_k b_k^n = \partial_t^m \partial_x^n f(x,0^+)$$