Nearby scalp channels in multi-channel EEG data exhibit high correlation. A question that naturally arises is whether it is required to record signals from all the electrodes in a group of closely spaced electrodes in a typical measurement setup. One could save on the number of channels that are recorded, if it were possible to reconstruct the omitted channels to the accuracy needed for identifying the relevant information (say, spectral content in the signal), required to carry out a preliminary diagnosis. We address this problem from a compressed sensing perspective and propose a measurement and reconstruction scheme. Working with publicly available EEG database, we put our scheme to experiment and illustrate that if it is only a matter of estimating the frequency content of the signal in various EEG bands, then all the channels need not be recorded. We have achieved an average error below 15% between the original and reconstructed signals with respect to estimation of the spectral content in the delta, theta and alpha bands. We have demonstrated that channels in the 10-10 system of electrode placement can be estimated, with an error less than 10% using recordings on the sparser 10-20 system.