Martingale Inequalities for Spline Sequences

A martingale is a sequence of values (w(n)) so that, provided one knows the first k values w(1),,w(k), the expectation of the (k+1)st value is precisely the kth value w(k). For instance, the sequence of winnings w(k) at time k in a fair game forms a martingale. Martingales are especially important in Probability Theory, but also in other branches of mathematics and physics (Brownian motion is also a martingale). There are many strong and useful theorems about martingales (of which many of them boil down to certain inequalities), exploiting their very special nature. We now consider sequences of spline functions (p(n)), which are (by definition) piecewise polynomial functions (i.e., they are piecewise sums of constant multiples of the functions 1,x,x^2,x^3 and so on) and additionally have some degree of smoothness at the breakpoints. For such sequences (p(n)), we introduce a concept of fairness which is similar to that for martingales. Splines are an important tool in Approximation Theory due to the fact that polynomials are among the most elementary functions and due to their smoothness properties. It turned out in recent years that many inequalities that are true for martingales can be transferred to fair spline sequences (p(n)), independently of the choice of breakpoints for the splines (p(n)) and in particular by literally stating the results without any additional assumption just replacing martingales by fair spline sequences. The intimate bond between martingales and splines was discovered only during the last 5 years by results developed in part by the author. The aim of the underlying project is to systematically transfer even more martingale results to the spline setting by exploiting and extending existing methods and thereby intensify the connection between martingales and fair spline sequences even further.

The concept of a fair game in probability theory has far-reaching applications, for instance in statistics, physics, or chemistry. Such fair games and their mathematical models are called "martingales". This rich concept is used in this project as a template for general qualitative and quantitative results about splines. Here, splines are piecewise polynomials having a certain smoothness and they are an important tool in approximation theory. The results obtained here lay the theoretical basis for fast convergence of adaptive spline algorithms and have applications for instance in the numerical solution of certain physical problems and in computer graphics.