Smooth Averaging and a Fourier Inequality

Consider a real-valued function over the integers, which could represent data collected at equally spaced time-intervals. It is often useful to smooth such a function by taking local weighted averages about each point. This raises the question: what is the best choice of such weights? In this talk, we investigate which choice of weights guarantees the smallest Laplacian of the smoothed function. Answering this question involves Fourier analysis, Chebyshev polynomials, and the equioscillation property.