کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6418055 | 1339319 | 2015 | 13 صفحه PDF | دانلود رایگان |

Let {λn}n=1â be a sequence of distinct complex numbers diverging to infinity so that |λn|â¤|λn+1| for all nâN, and let {μn}n=1â be a sequence of positive integers. Consider the setÎ:={λ1,λ1,â¦,λ1︸μ1-times,λ2,λ2,â¦,λ2︸μ2-times,â¦,λk,λk,â¦,λk︸μk-times,â¦}. Subject to the condition μnlogâ¡|λn|/|λn|â0 as nââ, we prove that all non-entire Taylor-Dirichlet series of the formân=1â(âk=0μnâ1cn,kzk)eλnz,cn,kâC, have a convex natural boundary if and only if Î is an interpolating variety for the space of entire functions of infraexponential type A|z|0. Our result is in the spirit of the Fabry-Pólya gap results.We also prove that if Î is the zero set of some FâA|z|0 but not an interpolating variety, it is still possible for the solutions of the differential equation of infinite order F(d/dz)f=0 to admit a Taylor-Dirichlet series representation, that is, a representation without groupings.
Journal: Journal of Mathematical Analysis and Applications - Volume 423, Issue 2, 15 March 2015, Pages 1825-1837