On the representation of infinitely differentiable functions by series of exponentials (Q1886105)

Let \(\psi:\mathbb R\longrightarrow[1,+\infty)\) and \(\alpha>1\) be such that (1) \(\exists_{A>0}:\;| \psi(x_1)-\psi(x_2)| \leq A(1+| x_1| +| x_2| )^{\alpha-1} | x_1-x_2| \), \(x_1,x_2\in\mathbb R\), (2) \(\lim_{x\to\infty}\psi(x)/| x| =+\infty\). Let \((M_n)_{n=0}^\infty\) be an increasing sequence with \(M_0=1\) such that: (3) \(M_n^2\leq M_{n-1}M_{n+1}\), \(n\in\mathbb N\), (4) \(\exists_{H_1,H_2>0}:\;M_n\geq H_1H_2^nn!\), \(n\in\mathbb Z_+\), (5) \(\forall_{s>1}:\;\liminf_{n\to+\infty}(M_{[sn]}/M_n^s)^{1/n}>1\), (6) \(\forall_{\delta>0}\;\exists_{p>0,\;t>1}:\;\sup_{m\in\mathbb N} \frac{M_{m+n}}{M_m(1+\delta)^m}\leq pt^nM_n\), \(n\in\mathbb Z_+\). Define \(\varphi(x):=\sup_{y\in\mathbb R}(xy-\psi(y))\), \(\theta_m(x):=\exp(\varphi(x)-m\log(1+| x| ))\), \(x\in\mathbb R\), \(m\in\mathbb N\). For \(\varepsilon_m\searrow0\), \(\sigma>0\), let \(G\) be the projective limit of the spaces \[ G_m:=\Big\{f\in\mathcal E(\mathbb R): \sup_{x\in\mathbb R,\;k\in\mathbb Z_+} \frac{| f^{(k)}(x)| }{(\sigma+\varepsilon_m)^kM_k\theta_m(x)}<+\infty\Big\},\quad m\in\mathbb N. \] The author studies the problem of expansion of functions from \(G\) in series of exponentials.