On the problem of mean periodic continuation
(I) Let \(T\in {\mathcal E}'(\mathbb{R}^n)\), \(B_{n,R}= \{x\in\mathbb{R}^n;|x|< R\}\), and \(B_{n,R}^T= \{x\in\mathbb{R}^n; x-y\in B_{n,R}\), for \(y\in \text{Supp\,}T\}\). Distributions from the class \[ D_T'(B_{n,R})= \{f\in D'(B_{n,R});\;f* T= 0\text{ on }B_{n,R}^T\} \] are said to be mean periodic on \(B_{n,R}\) with respect to \(T\). (II) For an entire function \(f: \mathbb C\to \mathbb C\), \(L(f)= \{z\in \mathbb C; f(z)=0\}\) and \(n_\lambda(f)\) is the multiplicity of \(\lambda\in L(f)\). \(\sigma_\lambda(f)= \sum_{j=0\sim n_\lambda-1}|a_{j,\lambda}(f)|\), \(a_{j,\lambda}(f)= (1/j!)((z- \lambda)^{n_\lambda}/f(z))^{(j)}|_{z\in\lambda}\). (III) Let \(M_+(\mathbb{R}^1)\) (resp., \(M_-(\mathbb{R}^1)\)) be the set of all \(T\in{\mathcal E}'(\mathbb{R}^1)\) (\(T\neq 0\), \(\widehat T:=\) Fourier transform of \(T\)) satisfying the following conditions: \(\sup_{\lambda\in L(\widehat T)}\text{Im\,}\lambda/\ln(2+|\lambda|)< \infty\) (resp., \(\text{inf}_{\lambda\in L(\widehat T)}\text{Im\,}\lambda/\ln(2+ |\lambda|)>-\infty\)), \(\sup_{\lambda\in L(\widehat T)}n_\lambda(\widehat T)/\ln(2+ |\lambda|)>\infty\), and \(\sigma_\lambda(\widehat T)\leq (2+ |\lambda|)^{\gamma_1}\), \(M(\mathbb{R}^1)= M_+(\mathbb{R}^1)\cap M_-(\mathbb{R}^1)\). Theorem 1 (due to one of the authors). Suppose that \(T\in M_+(\mathbb{R}^1)\) (resp., \(T\in M_-(\mathbb{R}^1)\)), \(\text{supp\,}T\supset[-r(T), r(T)]\), \(R> r(T)\), and \(f\in D_T'(B_{1,R})\). Then there exists a unique \(F\in D_T'(-\infty,R))\) (resp., \(F\in D_T'((-R,\infty))\)) for which \(F= f\) on \((-R, R)\). Here, both authors give the space \(M(\mathbb{R}^n\)) and the analogue of Theorem 1 for \(n\geq 2\) as Theorem 7 by using the spherical harmonics of degree \(k\) on \(S^{n-1}\). Finally, they give a condition of \(T\) under which there exists an \(f\in(\mathbb{C}^\infty_T\cap D_{k,j}')(B_{n,R})\), \(n\geq-2\), such that \(f\neq F|B_{n,R}\), whenever \(F\in D'(B_{n,R+\varepsilon})\) for some \(\varepsilon> 0\).
scientific article; zbMATH DE number 5666530