Approximation of the unit step function by a linear combination of exponential functions

AbstractWe approximate the unit step function, which equals 1 if t ε [0, T] and equals 0 if t >T, by functions of the form ∑n=1NAn(N)e−λnt/T, where each λn is a given positive constant. We find the coefficients An(N) by minimizing the integrated square of the difference between the unit step function and the approximating function. We first solve the specialized case where each λn = n. The resulting sum can be shown to converge in the mean to the unit step function as N → ∞. The general case is then solved and some interesting properties of the numbers An(N) are noted