I'm trying to solve problem 1.10 in Muscalu, Schlag, Classical and Multilinear Harmonic Analysis:
Show that for every $p>0$ there exists an approximate identity $K_{N,p}$ on $\mathbb{T}$ with the following properties:
- $\widehat{K_{N,p}}(\nu)=1$ for all $\lvert \nu\rvert\leq N$,
- $\widehat{K_{N,p}}(\nu)=1$ for all $\lvert \nu\rvert> CN$,
- $\lvert K_{N,p}(\theta)\rvert\leq C N^{1-p}\min(N^p, \lvert\theta\rvert^{-p})$,
where $C=C(p)$ is some constant and $N\geq 1$ is arbitrary.
First of all, I think that the second condition should read "$\widehat{K_{N,p}}(\nu)=0$ for all $\lvert \nu\rvert> CN$", otherwise condition 3 cannot be realized at $\theta=0$.
My idea was to exploit the fact that we already know that $(K_n)_{n\in\mathbb{N}}$ is an approximate identity, where $K_n(x)=\frac{1}{n}(\frac{\sin(N\pi x)}{\sin(\pi x)})^2$ are the Fejér kernels. Therefore I've tried to define
$K_{N,p}(\theta)=\frac{p+1}{p}K_{N(p+1)}(\theta)-\frac{1}{p}K_N(\theta)$.
With this definition, $(K_{N,p})_{N\in\mathbb{N}}$ is an approximate identity and satisfy the first two conditions (e.g. with $C(p)=p+1$), but I'm not able to verify the third one. If somebody has an idea how to prove or disprove it, or has another idea about how to define the approximate identity it would be really appreciated.