On convolution transforms whose inversion functions have complex roots

For convolution transforms it has been received inversion formula, when $\phi(x)=L^{2}(-\infty, +\infty)$, and inversion functions $E(s)=\prod\limits_{k=1}^{\infty}\Big(1-\dfrac{s^2}{a_k^2} \Big)$ have complex roots satisfying to conditions
$$\sum\limits_{k=1}^{\infty}<+\infty \dfrac {1}{|a _k|^2},~~|\arg a_k| \le \dfrac{\pi}{4}.$$