# How to interchange sum and integral?

We fix the point $$\xi_{0}\in \mathbb R.$$

Choose sequence $$\{f_{n}\}_{n\in \mathbb N}\subset L^{1}(\mathbb R)$$ with the following property :

(1) $$\|f_{n}\|_{L^{1}(\mathbb R)} \leq 1,$$ for $$n\in \mathbb N;$$ and (2) $$\hat{f_{n}}(\xi_{0})= 0,$$ for every $$n\in \mathbb N.$$

Define $$f:\mathbb R \to \mathbb C$$ as follows: $$f(x)=\sum_{n=1}^{\infty}\frac{2^{n-1}}{3^{n}} f_{n}(x), (x\in \mathbb R).$$

Since $$\|f_{n}\|_{L^{1}}\leq 1$$, and $$\sum_{n=1}^{\infty} \frac{2^{n-1}}{3^{n}}<\infty,$$ the series, $$\sum_{n=1}^{\infty}\frac{2^{n-1}}{3^{n}} f_{n}(x)$$ is absolutely summable in $$L^{1}(\mathbb R);$$ and hence, $$f\in L^{1}(\mathbb R).$$

My Question is: Can we expect $$\hat{f}(\xi_{0})= 0$$ ?

Ruff attempt: By definition of Fourier transform, we have,

$$\hat{f}(\xi_{0})= \int_{\mathbb R} f(x) e^{-2\pi i \xi_{0} \cdot x} dx = \int_{\mathbb R}(\sum_{n=1}^{\infty}\frac{2^{n-1}}{3^{n}} f_{n}(x)) e^{2\pi i \xi_{0}\cdot x} dx;$$ now from here, if I guess, we can interchange sum and integral; (but I don't know how to justify it); then by the hypothesis (2), $$\hat{f_{n}}(\xi_{0})=0;$$ it will follows that, $$\hat{f}(\xi_{0})=0.$$

• The series converges in $L^1$. The map $g \mapsto \int g(x)e^{2\pi i \xi\cdot x}\,dx$ is continuous for every $\xi$. That allows the interchange. – Daniel Fischer Jul 25 '14 at 15:46
• @DanielFischer; thanks, sorry I could not follow you, would you please tell me how does it help ? – Inquisitive Jul 25 '14 at 15:54
• Since the series converges in $L^1$, we have $$\int \left(\lim_{N\to\infty} \sum_{n=1}^N f_n(x)\right)e^{2\pi i\xi\cdot x}\,dx = \lim_{N\to\infty} \sum_{n=1}^N \int f_n(x)e^{2\pi i\xi \cdot x}\,dx$$ by continuity of the integral as a function of $g\in L^1$. – Daniel Fischer Jul 25 '14 at 16:02

The series converges in $L^1$. The map $g \mapsto \int g(x)e^{2\pi i \xi\cdot x}\,dx$ is continuous for every $\xi$. That allows the interchange. ...we have $$\int \left(\lim_{N\to\infty} \sum_{n=1}^N f_n(x)\right)e^{2\pi i\xi\cdot x}\,dx = \lim_{N\to\infty} \sum_{n=1}^N \int f_n(x)e^{2\pi i\xi \cdot x}\,dx$$ by continuity of the integral as a function of $g\in L^1$. -Daniel Fischer