APPROXIMATION ON A FINITE-SET OF POINTS THROUGH KRAVCHUK FUNCTIONS

In a harmonic oscillator environment, such as Fourier optics in a mult imodal parabolic index-profile fiber, data on a finite set of discrete observation points can be used to reconstruct the sampled wavefunctio n through partial wave synthesis of harmonic oscillator eigenfunctions . This procedure is generally far from optimal because a nondiagonal m atrix must be inverted. Here it is shown that Kravchuk orthogonal func tions (those obtained from Kravchuk polynomials by multiplication with the square root of the weight function) not only simplify the inversi on algorithm for the coefficients, but also have a well-defined analyt ical structure inside the measurement interval. They can be regarded a s the best set of approximants because, as the number of sampling poin ts increases, these expansions become the standard oscillator expansio n.