A representation of analytic functions in bounded convex domains in the complex plane

Entire functions of exponential type and regular growth are treated. Exceptional sets are investigated outside which these functions admit lower estimates that coincide asymptotically with upper estimates. An explicit method of constructing an exceptional set is indicated; the resulting set consists of disks with centers at zeros of the entire function in question. The concept of a properly balanced set is introduced, which is a natural generalization of the concept of a regular set by B. Levin. It is proved that of the zero set of an entire function is properly balanced if and only if every function, analytic in the interior of the conjugate diagram of the entire function in question and continuous up to the boundary is representable by a series of exponential monomials whose exponents are zeros of this entire function. This result generalizes the classical result of A. F. Leontiev on the representation of analytic functions in a convex domain to the case of where the entire function has multiple zeros.