Entire functions with fine asymptotic estimates for convex functions

In the paper we propose an entire function such that the logarithm of its modulus asymptotically approximates the given subharmonic function $\widetilde h(\operatorname{Re}z)$, where $\widetilde h$ is the Legendre transformation of a convex function $h(t)$ on $(-1;1)$ with the property $\exp(h(t))=o((1-|t|)^n)$, $n\in\mathbb N$. Such functions have applications in the issues on representation by exponential series of functions in integral weighted spaces on the interval $(-1;1)$ with the weight $\exp h(t)$. At that, better the approximation, a finer topology can be used for the representation by exponential series. For functions $h$ obeying $(1-|t|)^n=O(\exp(h(t)))$, $n\in\mathbb N$, the corresponding entire functions were constructed before. In the present paper we consider the functions satisfying $\exp(h(t))=o((1-|t|)^n)$, $n\in\mathbb N$. In the suggested construction we take into consideration the necessary conditions for the distribution of exponents for the exponentials in the unconditional bases obtained in previous works. This is why the main result of the paper (Theorem 1) should be treated not as a tool for constructing unconditional bases but as an argument supporting the absence of such bases.