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Saj_Eda
Saj_Eda
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Let the sequence $u_n\in L^2(0,\infty)$ weakly converges to $u\in L^2(0,\infty)$. What can we say about the corresponding Laplace transforms $u_n(s)$$U_n(s)$ and $u(s)$$U(s)$?

  1. $u_n(s)$$U_n(s)$ converges point-wise to $u(s)$$U(s)$ for almost all $s>0$.

  2. The convergence in (1) but also uniform.

Let the sequence $u_n\in L^2(0,\infty)$ weakly converges to $u\in L^2(0,\infty)$. What can we say about the corresponding Laplace transforms $u_n(s)$ and $u(s)$?

  1. $u_n(s)$ converges point-wise to $u(s)$ for almost all $s>0$.

  2. The convergence in (1) but also uniform.

Let the sequence $u_n\in L^2(0,\infty)$ weakly converges to $u\in L^2(0,\infty)$. What can we say about the corresponding Laplace transforms $U_n(s)$ and $U(s)$?

  1. $U_n(s)$ converges point-wise to $U(s)$ for almost all $s>0$.

  2. The convergence in (1) but also uniform.

Source Link
Saj_Eda
Saj_Eda
  • 395
  • 1
  • 11

Weak continuity under Laplace transform

Let the sequence $u_n\in L^2(0,\infty)$ weakly converges to $u\in L^2(0,\infty)$. What can we say about the corresponding Laplace transforms $u_n(s)$ and $u(s)$?

  1. $u_n(s)$ converges point-wise to $u(s)$ for almost all $s>0$.

  2. The convergence in (1) but also uniform.