### Mathematics Colloquium

# Peter Massopust

### (Helmholtz Zentrum München)

## "Splines of Complex Order, Fractional Differential Operators, and Dirichlet Averages"

** Date: ** |
Mon, October 24, 2011 |

** Time: ** |
17:15 |

** Place: ** |
Research II Lecture Hall |

**Abstract:** Splines of complex order $z$ are an extension of the classical polynomial Schoenberg splines $B_n$, $n\in \mathbb{N}$. Whereas the order $n$ for Schoenberg splines is only an indicator of the integral smoothness of $B_n$, the complex order $z$ provides a pair of indices with the real part representing the continuous smoothness class and the imaginery part the phase information. Thus, splines of complex order not only fill in the gaps between the integral smoothness classes but also provide a means of employing the phase information contained in $\Im z$ to resolve singularities such as edges and corners in images.
In this talk, we motivate the definition and present some of the basic theoretical background of splines of complex order, and indicate their relationship to fractional derivatives and integrals, and Dirichlet averages. In addition, we discuss potential applications of these splines to signal and image processing.