Analysis Problem of the Day 36

Today’s problem appears as Problem 5 on the Texas A&M August 2017 Real Analysis Qual:

Prove that for $a_1,...,a_N \in \mathbb{C},$

$\int_0^1 \left|\sum_{n=1}^N a_k e^{2\pi i k t}\right|^p dt \leq \sum_{k=1}^N |a_k|^p, \quad 1 \leq p \leq 2,$

and

$\int_0^1 \left|\sum_{n=1}^N a_k e^{2\pi i k t}\right|^p dt \geq \sum_{k=1}^N |a_k|^p, \quad 2 \leq p < \infty.$

Solution: The expressions appear to resemble Fourier series, so interpreting them as such, the statements reduce to showing that for 1-periodic functions $f,$ $\|f\|_p \leq \|\widehat{f}\|_{l^p}, 1 \leq p \leq 2,$ and $\|f\|_p \geq \|\widehat{f}\|_{l^p}, 2 \leq p < \infty.$ Now, recall that the Fourier transform $\mathcal{F}$ is a bounded operator $L^1 \to l^\infty, L^2 \to l^2,$ and similarly, its inverse $\mathcal{F}^{-1}$ is a bounded operator $l^1 \to L^\infty, l^2 \to L^2,$ with

$\|\mathcal{F}\|_{L^1 \to l^\infty}, \|\mathcal{F}\|_{L^2 \to l^2}, \|\mathcal{F}^{-1}\|_{l^1 \to L^\infty}, \|\mathcal{F}^{-1}\|_{l^2 \to L^2} \leq 1.$

By the Riesz-Thorin interpolation theorem, it follows the Fourier transform is bounded on $L^p \to l^{p'}, 1 \leq p \leq 2,$ and its inverse is bounded on $l^p \to L^{p'}, 2 \leq p' < \infty,$ with $\|\mathcal{F}\|_{L^p \to l^{p'}},\|\mathcal{F}^{-1}\|_{l^p \to L^{p'}} \leq 1,$ where $\frac{1}{p}+\frac{1}{p'}=1.$ Finally, recall that $\|a\|_{l^p} \leq \|a\|_{l^{p'}}, \|A\|_{L^{p'}([0,1])} \leq \|A\|_{L^p([0,1])}$ for $p' > p,$ i.e. the inclusions $l^p \hookrightarrow l^{p'}, L^{p'}([0,1]) \hookrightarrow L^{p}([0,1])$ are continuous with norm 1. It follows that $\mathcal{F}: L^{p'}([0,1]) \hookrightarrow L^p([0,1]) \to l^{p'}, 2 \leq p' \leq \infty$ and $\mathcal{F}^{-1}: l^p \to L^{p'}([0,1]) \hookrightarrow L^{p}([0,1]), 1 \leq p \leq 2$ are bounded operators with norm 1. Evaluating these expressions on functions with finite Fourier series yields the claim.

Leave a Reply Cancel reply