How we can reestablish communications with Pioneer 10?

It is very much possible for resumption of communications between base station and the satellite. It also is possible to re-establish communications with the previous satellites that left our solar system decades ago.
As it was explained before when the transmission reaches its destination the satellite is not at that place anymore and it has moved miles away from the previous place. Scientists do calculate the movements of Earth and satellite, the signal that is sent reaches it, The signal tells the computer on board to open memory and start recording the data; but by the time the transmission is over and the signal indicates that I am done and “end of transmission” the satellite is not there anymore. By referring to the formula, it shows there is 223 meters of frequency-Shift in every second of transmission to Pioneer 10, which is located only 12 hours (it is shown as 0.00137 of a light year) from Earth. (Click here to see the formula).

If a transmitter or a relay unit is aboard a plane or the transmission is sent to a relay satellite that travels at the speed of 500Km/h in the direction of east to west on the same latitude as Canberra, the frequency-shift is minimized by almost two third and is reduced to 89 meters, which is still receivable by the satellite. (Click here to see the formula). The transmissions are sent back by satellite can be received by the ground facilities as the returned signal is not subjected to a frequency-shift because the satellite is rotating like the Earth does. Corrections must be made to the way we transmit for a successful result in transmitting signals.
The Correction Transmitter can also be put on a train that travels at 150Km/h from east to west on a straight line. (Click here to see the formula). At this speed the frequency-shift is reduced to 176 meters from 223, at this rate we are still able to receive a fainted and distorted transmission. The transmission is sent from the ground station to the moving train, plain, or the stationery satellite and retransmitted to the satellite. The signal after received by the satellite is processed and sent back and 12 hours later is received by the ground station. The total time for the transmission would take anything more than 24 hours. This is important to know that there are other ways to reduce this frequency-shift even further, and it can be used re-establish communications with other satellites that previously lost communication with.

*Ask for details of these figures or see the formula.