I am reading a paper titled by Recovery of initial temperature from discrete sampling , and I have trouble with understanding the following:
We have two maps, the first is $a: F_r \rightarrow \mathbf{R^{n}}, $ where $F_r:=\{f \in W^{r,2}((0,\pi)): \sum_{k=1}^{\infty}k^{2r}|\hat{f_k}|^2 \leq 1\}$ where $\hat{f_k}$ is the Fourier sin coeffiecients, defined by $a(f)=(u_1,u_2,...,u_n)$, and $$u_i:=u_i(x_0,t_i)= \sum_{k=1}^{\infty}\hat{f_k}e^{-k^2t_i}sin(kx_o) \ \ \ \ \ \ \ \ \ \ ,with \ x_o \ irrational\ and\ (t_i)\ is\ increasing\ sequence. $$ The second is $M: \mathbf{R^n} \rightarrow L_2$.
They proved that $a$ is a continuous map, by letting $\ f,\ \bar{f}$ be two functions from $F_r$ and $\hat{f},\ \hat{\bar{f}},$ be their Fourier coefficients, then
My problem is with $$||\bar{f}-f||_{L_2}^{2}=\frac{\pi}{2}\sum_{k=1}^{\infty}|\hat{\bar{f}}-\hat{f}|\ \ \ \ \ (1)$$, they did not explain how they conclude that, I used the definition of the metric on $L_2$ space, but it does not help me.
Then, after this step they had $$|a(\bar{f})-a(f)|=|\bar{u}_j-u_j|\leq \sum_{k=1}^{\infty}|\hat{\bar{f}}-\hat{f}|e^{-k^2t_j} \leq \sqrt{\frac{2}{\pi}}||\bar{f}-f||_{L_2}\ ||(e^{-k^2t_1})_{k+1}^{\infty}||_{l_2}$$ which makes sence to me if I assume (1) holds. So I would appreciate any hint or help to explain 1.
Your formula (1) is missing an exponent: in the paper it is $$ ||\bar{f}-f||_{L_2}^{2}=\frac{\pi}{2}\sum_{k=1}^{\infty}|\hat{\bar{f}}-\hat{f}|^2 $$ This is just Parseval's identity: the normalizing coefficient $\pi/2$ appears because we are considering the sine series on $(0, \pi)$, where $$ \int_0^{\pi} (c_k \sin kx)^2\,dx = \frac{\pi}{2}c_k^2 $$
The second step you mentioned, $$ \sum_{k=1}^{\infty}|\hat{\bar{f}}-\hat{f}|e^{-k^2t_j} \leq \sqrt{\frac{2}{\pi}}||\bar{f}-f||_{L_2}\ ||(e^{-k^2t_1})_{k+1}^{\infty}||_{l_2} $$ involves the Cauchy-Schwarz inequality, so the terms $|\hat{\bar{f}}-\hat{f}|$ get squared.