Solution of the Ehrenpreis factorization problem. (English. Russian original) Zbl 0940.30030

Yulmukhametov, R. S.

Sb. Math. 190, No. 4, 597-629 (1999); translation from Mat. Sb. 190, No. 4, 123-157 (1999).

The author proves a general decomposition theorem for entire functions. As an application, using the Fourier-Laplace transform, he gives the full solution of the Ehrenpreis problem: For any \(\varphi\in\mathcal C_0^\infty(\mathbb R)\) with \(\operatorname{supp}\varphi\subset[-2N,2N]\), there exist \(\varphi_1, \varphi_2\in\mathcal C_0^\infty(\mathbb R)\) with \(\operatorname{supp}\varphi_{j}\subset[-N,N]\), \(j=1,2\), such that \(\varphi=\varphi_1\ast\varphi_2\). If, moreover, \(\varphi\) in a Gevrey class \(G_\alpha\), then \(\varphi_1\) and \(\varphi_2\) can be found in the same class \(G_\alpha\). If \(\mu\) is a distribution of finite order with \(\operatorname{supp} \mu\subset[-2N,2N]\), then there exist finite-order distributions \(\mu_1, \mu_2\) with \(\operatorname{supp}\mu_{j}\subset[-N,N]\), \(j=1,2\), such that \(\mu=\mu_1\ast\mu_2\).

Reviewer: M.Jarnicki (Kraków)

MSC:

30H05	Spaces of bounded analytic functions of one complex variable
30D15	Special classes of entire functions of one complex variable and growth estimates
46F10	Operations with distributions and generalized functions
42A85	Convolution, factorization for one variable harmonic analysis
32A15	Entire functions of several complex variables