A search for double beta processes in 190Pt and 198Pt was realized with the help of ultra-low background HP Ge 468 cm^3 gamma spectrometer in the underground Gran Sasso National Laboratories of the INFN (Italy). After 1815 h of data taking with 42.5 g platinum sample, T_{1/2} limits on 2beta processes in 190Pt (epsilonbeta^+ and 2epsilon) have been established on the level of 10^{14}-10^{16} yr, 3 to 4 orders of magnitude higher than those known previously. In particular, a possible resonant double electron capture in 190Pt was restricted on the level of 2.9 times 10^{16} yr at 90% C.L. In addition, T_{1/2} limit on 2 beta^- decay of 198Pt (2 u+0 u) to the 2^+_1 excited level of 198Hg has been set at the first time: T_{1/2}>3.5 times 10^{18} yr. The radiopurity level of the used platinum sample is reported.
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