Anharmonic approximation and bandlimited signals

The problem of approximating a given function (in the mean) on a finite interval by a finite sum of nonharmonic oscillations is solved by Hilbert space methods for the case in which the oscillation frequencies are prespecified. Such a problem arises in the mean square approximation of bandlimited signals, in determining the element amplitudes necessary to approximate a given radiation pattern for a linear antenna array with nonuniform element spacings, and in estimating the contribution of a particular reflector when a band-limited waveform is reflected from a set of fixed reflectors in space.It is shown that the general approximation problem can be reduced to that of solving the problem for a complex exponential function containing an arbitrary frequency parameter. An intimate connection is also exhibited between the mean square approximation of a bandlimited signal and the problem of interpolating a given set of data points by a bandlimited function of given bandwidth and minimum energy