Vladimir Temlyakov

Title:  Universality and Lebesgue inequalities in approximation and estimation


Abstract:

The concept of  the Kolmogorov width provides a very nice theoretical way of selecting an optimal approximation method. The major drawback of this way (from a  practical point of view)  is that in order to initialize such a procedure of selection we need to know a function class F. In many contemporary practical problems we have no idea which class to choose in place of F. There are two ways to overcome the above problem. The first one is to return (in spirit) to the classical setting that goes back to Chebyshev and Weierstrass. In this setting we fix a priori a form of an approximant   and look for an approximation method that is optimal or near optimal for each individual function from X. Also, we specify not only a form of an approximant but also choose a specific method of approximation (for instance, the one, which is known to be good in practical implementations). Now, we have a precise mathematical problem of studying efficiency of our specific method of approximation. This setting leads to the Lebesgue inequalities. The second way to overcome the mentioned above drawback of the method based on the concept of width consists in weakening an a priori assumption f in F. Instead of looking for an approximation method that is optimal (near optimal) for a given single class F we look for an approximation method that is near optimal for each class from a given collection F of classes. Such a method is called universal for F. We will discuss a realization of the above two ways in approximation and in estimation.