Локальні Властивості Гауссових Випадкових Полів На Компактних Симетричних Просторах І Теореми Типу Джексона Та Бернштейна

Abstract

We consider local properties of sample functions of Gaussian isotropic random fields on the compact Riemann symmetric spaces Yr rank one. We give conditions under which the sample functions of a field almost surely possess logarithmic and power modulus of continuity. As a corollary, we prove the Bernshtein-type theorem for optimal approximations 0 f functions of this sort by harmonic polynomials in the metric of space L 2 (Yd')" We use the Jackson-Bernshtein-type theorems to obtain sufficient conditions of almost surely belonging of the sample functions of a field to classes of functions associated with Riesz and Ces~ro means. Po3raa~amT~ca ~oKa.abni B.qaCTHBOCTi BI,16ipKOBI.lX d~yHKllJfl t"ayccOBrlX i30Tpommx BHna/~KOBI4X 110-YliB Ha KOMnalC.TltrlX piManOBnX CnMe'lprlqH14X npocTopax YV/ paltry 1. HaBe/~eHo yMOBI4, npl, l BrtKO-naHHi aKaX BH6ipKoBi qbyHKuii no~ta Malt>Ke ~mnemle MaloTb ~oraprlqbMimmlt ra ereneneBr~tt Mo/2y.ni nenepepBHocTi. J;IK Hac.ai~oK/loBeaello TeOpeMy THny l~epHm're~Ha ~t.qa OnTrlMaJIbl-ll4X na6.aa>KeHh TaKHx qbyHKtti~ I'apMOtliqnrlMl.I MllOrOq:lerlaMrl B MeTpmti npocTopy L 2 (Yv D. TeopeMa Trmy ~eKcona-BepHmTeitHa SaKopnc'TaHo/~Jl~ o'rprIMmma ]tOCTaTHiX yMOB Ha.aemHOCTi Maaace nanemle art-6ipKor~ax qbyHmti~ ~to KJmcin qbyuztti~t, nOB'X3aHHx 3 cepe~HiMu Picca Ta qe3apo.