use the short hand notation $$\label{011909}
{\frak s}(k):=\sin(\pi k)\quad {\frak c}(k):=\cos(\pi k),\quad k\in{{\mathbb T}}.$$ Let $
e_x(k):=\exp\{2\pi i xk\}$ for $x$ belonging to the set of integers ${{\mathbb Z}}.$ The Fourier series corresponding to a complex valued sequence $(f_x)_{x\in{{\mathbb Z}}}$ belonging to $\ell_2$ - the Hilbert space of square integrable sequences of complex numbers - is given by $$\label{fourier}
  \hat f(k)=\sum_{x\in{{\mathbb Z}}}f_xe_x^\star(k), \quad k\in{{\mathbb T}}.$$ Here $z^\star$ is the complex conjugate of $z\in\mathbb C$. By the Parseval identity $\hat f\in L^2({{\mathbb T}})$ - the space of complex valued, square integrable functions - and $\|\hat f\|_{L^2({{\mathbb T}})}=\|f\|_{\ell_2}$. Given ${\epsilon}>0$ we let ${{\mathbb Z}}_{{\epsilon}}:=({\epsilon}/2){{\mathbb Z}}$ and ${{\mathbb T}}_{\epsilon}:=(2/{\epsilon}){{\mathbb T}}$. Let $\ell_{2,{\epsilon}}$ be the space made of all complex valued square integrable sequences $(f_y)_{y\in
  {{\mathbb Z}}_{\epsilon}}$ equipped with the norm $$\|f\|_{\ell_{2,{\epsilon}}}:=\left\{\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}|f_y|^2\right\}^{1/2}.$$ Let $$\label{fouriereps}
  \hat f(\eta)=\frac{{\epsilon}}{2}\sum_{y\in{{\mathbb Z}}_{\epsilon}}f_ye_y^\star(\eta), \quad \eta\in{{\mathbb T}}_{\epsilon}.$$ The Parseval identity takes then the form $\|\hat f\|_{L^2({{\mathbb T}}_{\epsilon})}=\|f\|_{\ell_{2,{\epsilon}}}$.

For any non-negative functions $f,g$ acting on a set $A$ the notation $f\preceq g$ means that there exists a constant $C>0$ such that $f(a)\le Cg(a)$ for $a\in A$. We shall write $f\approx g$ if $f\preceq g$ and $g\preceq f$.

Given a function $f:\bar{{\mathbb R}}_+\to\mathbb C$ satisfying $|f(t)|\le
Ce^{Mt}$, for fome $C,M>0$ we denote 