Counstruct a sequence of Schwartz functions

Question

Here is an exercise from Wolff's lectures on harmonic analysis, p26:

Using translation and multiplication by characters, construct a sequence of Schwartz functions $\lbrace\phi_n\rbrace$ so that

1. Each $\phi_n$ has the same $L^p$ norm;
2. Each $\hat{\phi_n}$ has the same $L^{p'}$ norm;
3. The supports of the $\hat{\phi_n}$ are disjoint;
4. The supports of the $\phi_n$ are essentially disjoint, meaning that $$\Vert\sum_{n=1}^N\phi_n\Vert_p^p\approx\sum_{n=1}^N\Vert\phi_n\Vert_p^p$$uniformly in $N$.

I tried to countsruct the sequence as following: take a function $\phi\in\mathcal{S}$ such that $supp \hat{\phi}\subset B_1$, let $\hat{\phi_n}=\hat{\phi}(\cdot+2ne_1)$, then since $\phi_n(x)=e^{-4\pi inx_1}\phi(x)$, clearly $\lbrace\phi_n\rbrace$ satisfies 1-3. However, I have trouble in proving 4: now $$\int\vert\sum_{n=1}^Ne^{-4\pi inx_1}\phi(x)\vert^pdx=\int\vert\frac{\sin 2\pi nx_1}{\sin 2\pi x_1}\phi(x)\vert^pdx$$I do not know how to show this integral is approximately $N$. Also, I do not know whether I got a suitble construction of the sequence.

Could anyone tell me how to estimate the last integral or provide an appropriate construction? Any help will be highly appreciated!

Answer 1

No response :( But now I know how to solve it.

First we need a lemma:

Lemma. Let $1\leqslant p<\infty$. Suppose $f,g\in L^p(\mathbb{R}^d)$, then $$\lim_{h\rightarrow\infty}\Vert f+g(\cdot+h)\Vert_p^p=\Vert f\Vert_p^p+\Vert g\Vert^p_p$$

The lemma is a easy consequnce of the denseness of $C_c(\mathbb{R}^d)$ in $L^p(\mathbb{R}^d)$.

Now back to our problem. Choose a function $\phi\in\mathcal{S}$ such that $supp\hat{\phi}\subset B_1$, let $$\hat{\phi}_n(\xi)=e^{2pia_n\xi_d}\hat{\phi}(\xi+2ne_1)$$ where $a_n$ to be determined. Clearly condition 1.2.3. are satisfied for arbitrary $a_n$. Now, using the lemma stated above we can choose $a_n$ sufficiently large such that $$(1-\frac{1}{2^N})(\Vert\phi_N\Vert^p_p+\sum^{N-1}_{n=1}\Vert\phi_n\Vert_p^p)\leqslant\Vert \phi_N+\sum_{n=1}^{N-1}\phi_n\Vert^p_p\leqslant(1+\frac{1}{2^N})(\Vert\phi_N\Vert^p_p+\sum^{N-1}_{n=1}\Vert\phi_n\Vert_p^p)$$ by induction. Then 4. is satisfied.