Definition:Convergent Sequence/Test Function Space

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Definition

Let $\map \DD {\R^d}$ be the test function space with the compact support $K \subseteq \R^d$.

Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence in $\map \DD {\R^d}$.

Let $\phi \in \map \DD {\R^d}$ be a test function.

Let $D^k := \dfrac {\partial^{k_1 + k_2 + \ldots + k_d}} {\partial x_1^{k_1} \partial x_2^{k_2} \ldots \partial x_d^{k_d} }$ be a partial differential operator with the multiindex $k = \tuple {k_1, k_2, \ldots, k_d}$.

Suppose:

$\forall n \in \N : \forall x \in \R^d \setminus K : \map {\phi_n} x = 0$

Suppose $\sequence {\phi_n}_{n \mathop \in \N}$ converges uniformly to $\phi$.

Suppose that for every multiindex $k$ the sequence $\sequence {D^k \phi_n}_{n \mathop \in \N}$ converges uniformly to $D^k \phi$.


Then the sequence $\sequence {\phi_n}_{n \mathop \in \N}$ converges to $\phi$ in $\map \DD {\R^d}$.


This can be denoted:

$\phi_n \stackrel \DD {\longrightarrow} \phi$


Further research is required in order to fill out the details.
In particular: Check how to make the partial differential operator work with complex variables
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Also see

  • Results about convergent sequences in test function space can be found here.


Sources

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