Interchange of sums and integral

Question

Suppose I have a function $x:\mathbb{R}\rightarrow\mathbb{R}$ such that is square-integrable:

$$\int_{-\infty}^\infty|x(t)|^2dt<\infty$$

Suppose also that $x(t)$ contains no higher frequencies than $B$ Hz. Then I can use the Nyquist theorem to recover $x(t)$ from its samples as long as these samples are spaced $T_s=\frac{1}{2B}$ seconds apart. I am trying to prove that the sequence $\{x(kT_s)\}_{k=-\infty}^\infty$ of samples of $x(t)$ is square-summable. Here is my approach:

$$\begin{eqnarray} \infty&>&\int_{-\infty}^\infty|x(t)|^2dt\\ &=&\int_{-\infty}^\infty\left(\sum_{k=-\infty}^\infty x(kT_s)\operatorname{sinc}(t/T_s-k)\right)\left(\sum_{l=-\infty}^\infty x(lT_s)\operatorname{sinc}(t/T_s-l)\right)dt\\ &=&\sum_{k=-\infty}^\infty \sum_{l=-\infty}^\infty x(kT_s)x(lT_s)\int_{-\infty}^\infty\operatorname{sinc}(t/T_s-k)\operatorname{sinc}(t/T_s-l)dt\\ &=&T_s^2\sum_{k=-\infty}^\infty |x(kT_s)|^2 \end{eqnarray}$$

However, I am unsure about the third step. Can I interchange the sums with integral there? Does Fubini's Theorem apply here?

Answer 1

Since the Integral of the sums converges, Then Fubini's theorem guarantees that the sum of the integral will also converge to the same answer.

A similar question was answered here.

Answer 2

Partial answer: define suitable parameters $t$ in a suitable metrics spaces $T$ and $Y$ in theorem.

Theorem. Let $\{ F_t ; t\in T\}$ be a family of functions $F_t : Y \rightarrow \mathbb{C}$ depending on a parameter t; let $\mathcal{B}_X$ be a base $Y$ and $\mathcal{B}_{T}$ a base in $T$. If the family converges uniformly on $Y$ over the base $\mathcal{B}_{T}$ to a function $F : Y \rightarrow \mathbb{C}$ and the limit $\lim_{\mathcal{B}_{T}} F_t(y)=A_t$ exists for each $t\in T$, the both repeated limits $\lim_{\mathcal{B}_{Y}}(\lim_{\mathcal{B}_{T}}F_t(y))$ and $\lim_{\mathcal{B}_{T}}(\lim_{\mathcal{B}_{Y}}F_t(y))$ exist and the equality and holds $$ \lim_{\mathcal{B}_{Y}}(\lim_{\mathcal{B}_{T}}F_t(y))=\lim_{\mathcal{B}_{T}}(\lim_{\mathcal{B}_{Y}}F_t(y)). $$

This theorem can be found in books of Zorich (Mathematical Analysis II p. 381). In this case, if the integral $\displaystyle\lim_{t\to\infty}\int_{-t}^t|x(t)|^2dt$ or the sequence $$ \lim_{y\to\infty}\left(\sum_{k=-y}^y x(kT_s)\operatorname{sinc}(t/T_s-k)\right)\left(\sum_{l=-y}^y x(lT_s)\operatorname{sinc}(t/T_s-l)\right) $$ converge uniformly com respect to $y$ or $t$ in this order, then set $$ F_t(y)= \int_{-t}^t\left(\sum_{k=-y}^y x(kT_s)\operatorname{sinc}(t/T_s-k)\right)\left(\sum_{l=-y}^y x(lT_s)\operatorname{sinc}(t/T_s-l)\right)dt $$ and use the Theorem.