REPRESENTATION OF AN ENTIRE FUNCTION AS A PRODUCT OF 2 FUNCTIONS OF EQUIVALENT GROWTH

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 compac t support in R(n) (n greater than or equal to 2) can be represented as a convolution of two smooth functions with compact support. Dickson p roved that a smooth function of one variable with compact support can be represented as a convolution of two smooth functions with compact s upport if all the zeros lambda(k) of the Fourier-Laplace transform of this function are located in some horizontal strip (\lambda k\less tha n or equal to r) Sigma 1 = Dr + O(1) as r --> infinity. It is proved i n the present paper that the factorization is possible if all the zero s of the Fourier-Laplace transform are located in a domain of the foll owing form: G(a) = {z = x + iy, \y\ less than or equal to exp(a root l n(\x\ + 1))}.