On the Rate of Best Approximation

Start: 02/10/2010 - 4:15pm
End  : 02/10/2010 - 5:15pm


Professor Timur Oikhberg, University of California Irvine


Determining the rate of best approximation of a given function by functions of a certain class (such as algebraic or trigonometric polynomials of fixed degree) is one of the main topics of approximation theory. The classical theorems of Bernstein and Jackson establish the connection between the rate of approximation, and the smoothness of a function. With no smoothness assumptions, no information about the rate of approximation by polynomials can be extracted. Suppose now our class of approximating functions is richer than the set of polynomials (think, for instance, of piecewise polynomial functions, or splines). We consider whether the rate of best approximation with respect to such a system can be arbitrarily slow. It turns out that, in many cases, the answer is positive: if (a_n) is a sequence of positive numbers deceasing to 0, then one can find a function f such that the distance from f to the nearest member of the approximating class is at least a_n (for many different methods of calculating the distance).

Time permitting, we will also consider the related problem of approximating matrices (or operators) by matrices of prescribed rank.

Beckman B126, Harvey Mudd College

Misc. Information

Refreshments will be served at 3:45 pm in Olin B 161, Harvey Mudd College. The dinner will be hosted by Asuman Akson. If interested in attending, please call x72769.