"Interpolation" simply means finding a function g from a given class of functions (e.g. polynomials) that runs through given (x, y) points in the plane. This method naturally invites you to approximate a given (generally complicated) function f by a function g from a class of much simpler functions - simply by evaluating the function f at a finite number of points. The "size" of the difference between f and g (the approximation error) depends crucially on the class of functions used for interpolation. We will interpolate with polynomials and so-called radial basis functions and compare the approximation errors with each other.

One tends to think that the approximation error approaches zero if the number of points at which the function f is evaluated approaches infinity. We will show that this is a fallacy. Even "the opposite" is the case: For certain, so-called "universal functions" f, you can find a subsequence of interpolants for every (suitable) function h that converges towards h - and therefore not towards f.

Another way to approximate data or functions is the so-called Fourier analysis - the decomposition of functions into their individual frequencies. We will look at the basic principles and ideas of this theory as well as the many possible applications, e.g. in audio or image analysis.

Finally, we will look at many small mathematical curiosities and bizarre facts from various areas of mathematics that can easily be used in the classroom, such as the question of how two people can share a pizza fairly.

Program