UAlberta-PIMS Mathematics and Statistics Colloquium: Stefano De Marchi

In this talk, we start by recalling the well-known problem of multivariate approximation by polynomials of total degree, concerning finding good points for interpolation. This allowed the discovery of the Padua Points, the first set of unisolvent and explicitly known points on the square whose Lebesgue constant has optimal growth. i.e. O(logˆ2(n)), with n the polynomial degree. A more general set is the points obtained by sampling on Lissajous curves, which have the advantage of being defined in all space dimensions and have found application in Magnetic Particle Imaging, an emerging medical imaging technique alternative to MRI. Different approaches when approximating functions (even discontinuous). We describe the Variably Scaled (Discontinuous) Kernels and their suitability in approximate multivariate functions. Finally, a general approach called the ``fake” nodes approach, which is essentially a mapping basis technique, will be described.