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On the solvability of the Cauchy-Riemann equation with constraints of the growth of the functions and weighted approximation of analytic functions. (Russian) Zbl 0696.30041
Let $$\phi$$ be a subharmonic function in a domain $$\Omega$$ such that $$\phi =\sup \{\log | f|:$$ f is holomorphic in $$\Omega$$ and log$$| f| \leq \phi$$ in $$\Omega$$ $$\}$$. Let g be a measurable function such that $$| g(z)| \leq e^{\phi (z)}/| z|$$ in $$\Omega$$. Then there exists a measurable function f such that $| f(z)| \leq c(1+| \log | z| |)e^{\phi (z)}$ on $$\Omega$$ and $$\partial f/\partial \bar z=g$$ on $$\Omega$$ in the distributional sense, where c is a constant. The theorem is applied to the problem of approximation of analytic functions of a given growth by analytic functions of a smaller growth.
Reviewer: J.Siciak

##### MSC:
 30E99 Miscellaneous topics of analysis in the complex plane 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
##### Keywords:
subharmonic function