Henri Padé wrote his PhD thesis in 1867 at the École Normale Supérieure in Paris. The dissertation was on what we know today as the Pade approximant. Come to think if it today, it is really remarkable that 100 years ago, mathematicians had thought about function approximations using rational polynomials. Now it may appear all too simple, but without the aid of a serious computing machine, one would have to rely on the mathematical rigour on every small argument. In comparison, these days, we can quickly check the validity using some computer program before venturing into a formal proof.
Anyway, the sudden recollection of this French mathematician happened, as I was looking for a good curve fit for a problem I need as part of my work. I needed to minimize the maximum error than the average or squared error. I thought about Legendre polynomial fit, which gave me the minimum root mean squared error and Chebychev who minimized the worst case error. The order of the polynomial and the number of sample points seemed to have dependency. Pade approximant is a cool technique which reduces the polynomial order while using a rational polynomial. I am not too convinced about the statistical properties of this method. The data points I have at disposal may have some measurement error attributed. I need to investigate a bit more before taking a decision on the potential optimality for the given problem! Happy knowing this scheme though. In retrospect, I remember hearing it in my statistical signal processing books, but never paid any serious attention then! Silly.