Optimum impulse response and the van der Maas function

The optimum impulse response of a band-limited system, viz., the van der Maas function, is derived from considerations based on the theory of entire functions. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">²</tex> version of the optimization problem is also discussed. In particular, it is shown that there is no square integrable impulse response that is optimum in Chebyshev's sense since the van der Maas function, which is not square integrable, can be regarded as the limit of a sequence of square integrable functions. Some modified <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L²</tex> versions of the optimization problem, in which weighted square integral measures of the sidelobes are prescribed, are also described.