Inequalities on time-concentrated or frequency-concentrated functions

We obtain an inequality on a measure of the spread in time of periodic functions that are concentrated in frequency, i.e. all but a fixed finite number of Fourier coefficients vanish with meansquared error up to . We characterize an extremal function and give an asymptotic formula for the measure of spread of this extremal function as approaches 0. We also consider the corresponding problem for functions on the real line that are -concentrated in time or frequency. When = 0, the above reduce to inequalities on time-limited or band-limited functions and these are discussed in more detail.