Fourier coefficients of Selberg polynomials

Question

In Montgomery's "Ten Lectures on the Interface Between Number Theory and Harmonic Analysis" a bound for the Fourier coefficients of the Selberg polynomial $S^+_K$ is obtained by using what he calls Vaaler's lemma.

I am interested in obtaining a similar bound directly without using Vaaler's lemma (Montgomery states "We now require an estimate for $|\hat{S}_K^+(k)|$. Since $S_K^+(x)$ has been explicitly defined, one might argue directly, but ..." so it seems possible.) The reason being I want to bound the Fourier coefficients of something similar to Selberg polynomial (defined below) where I can not apply Vaaler's lemma as in the referenced book. So I wanted to understand how I can do this for Selberg polynomial first.

However, I am not getting anything close to the bound $$ |\hat{S}_K^+(k)| \leq \frac{1}{K+1} + \min (\beta - \alpha, \frac{1}{\pi |k|}) $$ so far.

I apologize this is a technical question, but if someone could get me started on this I would greatly appreciate it.

Definitions: Let $K$ be a positive integer. Vaaler's polynomial is defined $$ V_K(x) = \frac{1}{K+1}\sum_{k=1}^K\left(\frac{k}{K+1} - \frac12\right) \Delta_{K+1}\left(x - \frac{k}{K+1}\right) + \frac{1}{2 \pi (K+1)}\sin 2 \pi (K+1) x - \frac{1}{2 \pi} \Delta_{K+1}(x) \sin 2 \pi x, $$ where $$ \Delta_K(x) = \sum_{-K}^K \left(1 - \frac{|k|}{K}\right)e(kx) $$ is the Fejér kernel. Then the Selberg polynomial for the interval $[\alpha, \beta]$ is defined $$ S^+(x) = \beta - \alpha + B_K(x - \beta) + B_K( \alpha - x) $$ where $$ B_K (x) = V_K(x) + \frac{1}{2 (K+1)} \Delta_{K+1}(x). $$

PS my previous related question Showing Vaaler polynomial is a good approximation to saw tooth function was an attempt to do this...

Answer 1

As it is odd, we can write the Vaaler polynomial $V_K(x)=\sum_{1 \le k \le K}c_{k,K} \sin 2\pi kx$ and its fundamental property is that $|c_{k,K}| \le \frac{1}{\pi k}$.

This follows easily from its definition in Vaaler's paper referenced before (there it is called $\psi*j_N$, where $j_N$ is defined on page $207$ and its properties are spelled out in Theorem $18$ on page $210$

(odd follows from $7.13$ in the paper as it has the sign of $\psi$ the sawtooth function and the bound because $|\hat J_{N+1}(n)|= |\hat J(\frac{n}{N+1})| \le 1$ and $\psi*j_N(x)=-\sum_{n=-N, n \ne 0}^N(-2\pi i n)^{-1}\hat J_{N+1}(n)e(nx)$, where $J, \hat J$, defined on page $191$ are analyzed in Theorem $6$ page $192$ )

Then $V_K(x - \beta) + V_K( \alpha - x)=\sum_{1 \le k \le K}c_{k,K} (\sin 2\pi k(x-\beta)+\sin 2\pi k(\alpha-x))$, so

$ V_K(x - \beta) + V_K( \alpha - x)=\sum_{1 \le k \le K}(2c_{k,K}\sin \pi k(\alpha-\beta))\cos (\pi k (2x-\alpha-\beta))$ and clearly the latter written as Fourier series (so the cosine term is expanded as the half sum of two exponentials) has coefficients at most $|c_{k,K}\sin \pi k(\alpha-\beta))| \le \min (|c_{k,K}|, |c_{k,K}(\pi k (\beta-\alpha)| $ in absolute value and we have that being at most $\min (\beta - \alpha, \frac{1}{\pi |k|})$ by the above

The other part in the $B_K$ is the sum of two terms, each being $ \frac{1}{2 (K+1)}$ times a Fourier series with coefficients at most $1$ in absolute value, hence the $\frac{1}{K+1}$ term.

If $k \ne 0$ we are done by the above as we indeed get $|\hat{S}_K^+(k)| \leq \frac{1}{K+1} + \min (\beta - \alpha, \frac{1}{\pi |k|})$, while if $k=0$ the Vaaler polynomial term disappears and the inequality is trivial and is an equality since the $0$ th terms of $\Delta_{K+1}(x-\beta), \Delta_{K+1}(\alpha-x)$ are both $1$.