Find sequence of functions $f_n \in C_0^\infty (-1,1)$ converging to $0$ in $L^2$ but $f_n(0) \rightarrow \infty$

Question

Are there some nice examples for sequences of functions $f_n \in C_0^\infty(-1,1)$ converging to $0$ in $L^2$ but $f_n(0) \rightarrow \infty$ ?

Answer 1

The following does not depend on $n$: $$ \int_{-\infty}^{\infty}e^{-x^2}dx=\int_{-\infty}^{\infty}ne^{-(ny)^2}dy $$ So $f_n=\sqrt{n}e^{-(ny)^2/2}$ defines a sequence $\{ f_n \}_{n=1}^{\infty}$ of unit vectors in $L^2(\mathbb{R})$. Therefore, $f_n(x)=n^{1/4}e^{-(ny)^2/2}$ tends to $0$ in $L^2(\mathbb{R})$ as $n\rightarrow\infty$, even though $f_n(0)\rightarrow\infty$.

It's not hard to modify the argument starting with a function $g(x) \in C_c^\infty$ instead of $e^{-x^2}$.