Approximation of Müntz–Szasz type in weighted $L^p$ spaces, and the zeros of functions of the Bergman classes in a half-plane

We study the completeness of the system of exponents $\exp(-\lambda_nt)$, $\operatorname{Re}\lambda_n>0$, in spaces $L^p$ with the power weigh on the semiaxis $\mathbb R_+$. We prove a sufficient condition for the completeness; one can treat it as a modification of the well-known Szasz condition. With $p=2$ it is unimprovable (in a sense). The proof is based on the results (which are also obtained in this paper) on the distribution of zeros of functions of the Bergman classes in a halfplane.