Convergence in Spaces of Rapidly Increasing Distributions

Saleh Abdullah

Abstract: In this note we show that if $(T_j)$ is a sequence in $K'_M$, the space of distributions of rapid growth (resp. $O'_c$ the space of its convolution operators), and $(T_j \star \phi)$ converges to 0 in $K'_M$ (resp. in $O'_c$) for all $\phi$ in $K_m$, then $(T_j)$ converges to 0 in $K'_M$ (resp. $O'_c$). Moreover, if $(\psi_j)$ is in $O_c$ such that $(\psi_j \star \phi)$ converges to 0 in $O_c$ for every $\phi$ in $K_M$, then $(\psi_j)$ converges to 0 in $O_c$. This is no more true if the sequence $(\psi_j)$ is in $K_M$.

Classification (MSC2000): 46F05, 46F10

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