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.