Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University3469510.14498/vsgtu1139Research ArticleThe Dunkl convolution operators and multipoint de la Vallee–Poussin problemZabirovaKarina Raisovnakarinazabirova@gmail.comNapalkovValentin Vasil'evichDoctor of physico-mathematical sciences, Professorshaig@anrb.ru, napalkov@matem.anrb.ru, vnap@matem.anrb.ruUfa State Aviation Technical UniversityInstitute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences15122013171708110062020Copyright © 2013, Samara State Technical University2013The Dunkl operator as an object of mathematical physics is considered, we study the kernel and the surjectivity of Dunkl convolution operators in the space of entire functions and the space of entire functions of exponential type. The main result is the solution of the multipoint de la Vallee–Poussin problem for Dunkl convolution operators in the space of entire functions.Dunkl operatorsDunkl convolutionde la Vallee–Poussin (Cauchy) problemsufficient setsspace of entire functionsоператор Данкласвёртка Данклазадача Валле–Пуссена (Коши)достаточные множествапространство целых функций[L. Lapointe, L. Vinet, "Exact operator solution of the Calogero-Sutherland model", Commun. Math. Phys., 178:2 (1996), 425-452][J. J. Betancor, M. Sifi, K. Trimeche, "Hypercyclic and chaotic convolution operators associated with the Dunkl operators on ", Acta Math. Hung., 106:1-2 (2005), 101-116][В. В. Напалков, В. В. Напалков (мл.), "Операторы Данкла как операторы свертки", Докл. Акад. наук, 423:3 (2008), 300-302][В. В. Напалков, Уравнения свертки в многомерных пространствах, Наука, М., 1982, 240 с.][J. Dieudonne, L. Schwartz, "La dualite dans les espaces () et ()", Ann. Inst. Fourier, 1 (1949), 61-101][L. Euler, "De integratione aequationum differentialum altiorum gradum", Miscellanea Berol., 7 (1743), 193–242][H. S. Shapiro, "An algebraic theorem of Fisher, and the holomorphic Goursat problem", Bull. Lond. Math. Soc., 21:6 (1989), 513-537][В. В. Напалков, "О строгой топологии в некоторых весовых пространствах функций", Матем. заметки, 39:4 (1986), 529-538][О. В. Епифанов, "О существовании непрерывного правого обратного в одном классе локально выпуклых пространств", Изв. Сев.-Кавк. научн. центра высш. шк. Естеств. науки, 1991, № 3(75), 3-4]