Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?

Can one derive results about entire function of exponential type by using results about exponential polynomials ?

For example I am wondering if it is possible to derive sampling theorems about band-limited functions by studying properties of exponential polynomials ?

What about the distribution of zeros ?


Exponential polynomials are very special among the entire functions of exponential type. Their zeros accumulate to finitely many directions: there are finitely many numbers $\{\theta_1,\ldots,\theta_n\}$ such that the arguments of zeros accumulate only to those $\theta_j$. General entire functions of exponential type can have arguments of zeros arbitrarily distributed, for example, uniformly.

Further, $\log M(r)$, where $M(r)=\max_{|z|\leq r}|f(z)|$ for exponential polynomials is like $(c+o(1))r$, while for general entire functions of exponential type, the limit $(\log M(r))/r$ might not exist.

There are many other properties specific to exponential polynomials: their Taylor coefficients, zeros and everything else.


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