Series of exponentials in the space of sequences of finite order and type. (Q1432290)

Let \(l_\infty(\rho,\sigma)\) be a Banach space of sequences \(x= (x_n)^\infty_{n=-\infty}(x_n\in\mathbb{C})\) with the norm \(\| x\|= \sup_n\{| x_n|/ \exp( \sigma| n|^\rho)\), where \(\rho >1\), \(\sigma>0\) and \(n\neq 0\). Let \(A(\rho, \sigma)\) be the projective limit of the spaces \(l_\infty(\rho,\sigma +\varepsilon) (\varepsilon>0)\). Let the sequence \((\lambda_k)^\infty_{k=1}\) be contained in \(\{z:\text{Im}\,z\in[0,2\pi)\}\) and \(|\lambda_k |\uparrow\). Construct a \(2\pi i\)-periodic entire function \(G(\lambda)\) with simple zeros \(\lambda_k\) \((k=1,2,\dots)\) and verifying some other conditions. Let \(d^n_k\) be the \(n\)-th Fourier coefficient of \(G(\lambda)/(e^{\lambda-\lambda_k}-1)\). Then \(x\in A(\rho,\sigma_1)\) \((0<\sigma_1<\sigma)\) can be represented by a series absolutely convergent in the topology of \(A(\rho,\sigma)\) as follows; \(x= \sum^\infty_{k=1} \beta_k \exp^{\lambda_k}\), where \(\beta_k=\sum^\infty_{n=-\infty}x_n (d^k_n/G' (\lambda_k))\) and \(\exp^{\lambda_k} (n\lambda_k))^\infty_{n=-\infty}\).