Representation of an~entire function as a~product of two functions of equivalent growth

R. S. Yulmukhametov

Sbornik. Mathematics, Tome 187 (1996) no. 7, pp. 1087-1107

Résumé

A problem of Ehrenpreis on factorization in the convolution algebra of smooth functions with compact support is considered. It was proved at the beginning of the 1980s that not every smooth function with compact support in $\mathbb R^n$ ($n\geqslant 2$) can be represented as a convolution of two smooth functions with compact support. Dickson proved that a smooth function of one variable with compact support can be represented as a convolution of two smooth functions with compact support if all the zeros $\lambda _k$ of the Fourier–Laplace transform of this function are located in some horizontal strip and $$ \sum _{|\lambda _k|\leqslant r}1=Dr+O(1)\qquad \text{as }\ r\to \infty. $$ It is proved in the present paper that the factorization is possible if all the zeros of the Fourier–Laplace transform are located in a domain of the following form: $$ G_a=\bigl \{z=x+iy,\ |y|\leqslant \exp \bigl (a\sqrt {\ln (|x|+1)}\,\bigr)\bigr \}. $$