B737 Captain reported failure with the autopilot; autothrottle; GPS; and FMS in an area of reported GPS jamming. The autopilot eventually returned after following procedures but the same problem occurred again 30 minutes later.

Narrative

At cruise. We were told that there was GPS jamming testing in progress. Moments later we received an Multipurpose Control Display Unit (MCDU) message that a GPS was offline. Shortly after that we lost all FMS data - screens went blank; loss of Enhanced Ground Proximity Warning System (EGPWS) info; loss of both autopilots and loss of autothrottles. Also momentary flashing of flight control. I reverted to hand-flying; alerted ATC of our issue and asked to be vectored; I also told them we could not maintain RVSM (Reduced Vertical Separation Minimum) without a working autopilot. I transferred the controls to the FP (PM); set up for manual and ground-based green needle navigation; verified full manual control; and ran through the FMS system loss QRH. We then went through the FMS reestablish QRH. We were able to bring them both back online but no autopilot until we ran through the FMS reestablish procedure. As we were descending as cleared out of RVSM; we were able get the autopilot back and we were cleared to return to our planned altitude. After about another 30 minutes of normal operation; the same issue happened; in pretty much the same order; and we were able to get everything back online within 2 minutes.