On the rate of decay of a Meyer scaling function

A function with the following properties is called a Meyer scaling function: $\varphi\colon\Bbb R\to\Bbb R$, its integral shifts $\varphi(\cdot+n)$, $n\in\Bbb Z$, are orthonormal in $L_2(\Bbb R)$, and its Fourier transform $\widehat{\varphi}(y)=\frac{1}{\sqrt{2\pi}}\int\limits_{\Bbb R}\varphi(t)e^{-iyt} dt$ has the following properties: $\widehat{\varphi}$ is even, $\widehat{\varphi}=0$ outside $[-\pi-\varepsilon,\pi+\varepsilon]$, $\widehat{\varphi}=\frac{1}{\sqrt{2\pi}}$ on $[-\pi+\varepsilon,\pi-\varepsilon]$, where $\varepsilon\in\bigl(0,\frac{\pi}{3}\bigr]$. Here is the main result of the paper. Assume that
$$ \omega\colon[ 0,+\infty)\to [ 0,+\infty) $$
and the function $\frac{\omega(x)}{x}$ decreases. Then the following assertions are equivalent. 1. For every (or, equivalently, for some) $\varepsilon\in(0,\frac{\pi}{3}]$ there exists $x_0>0$ and a Meyer scaling function $\varphi$ such that $\widehat{\varphi}=0$ outside $[-\pi-\varepsilon,\pi+\varepsilon]$ and $|\varphi(x)|\leqslant e^{-\omega(|x|)}$ for all $|x|>x_0$. 2. $\int\limits_1^{+\infty}\frac{\omega(x)}{x^2} dx<+\infty$.